Question 1

Two triangles can be formed using the given measurements. Solve both triangles.
- B=35∘,b=23,c=24\mathrm { B } = 35 ^ { \circ } , \mathrm { b } = 23 , \mathrm { c } = 24B=35∘,b=23,c=24

Accepted Answer

A)    A=91.8∘,C=53.2∘,a=13.2;A=88.2∘,C=126.8∘,a=13.2\mathrm { A } = 91.8 ^ { \circ } , \mathrm { C } = 53.2 ^ { \circ } , \mathrm { a } = 13.2 ; \mathrm { A } = 88.2 ^ { \circ } , \mathrm { C } = 126.8 ^ { \circ } , \mathrm { a } = 13.2A=91.8∘,C=53.2∘,a=13.2;A=88.2∘,C=126.8∘,a=13.2 
B)    A=91.8∘,C=53.2∘,a=40.1;A=88.2∘,C=126.8∘,a=40.1\mathrm { A } = 91.8 ^ { \circ } , \mathrm { C } = 53.2 ^ { \circ } , \mathrm { a } = 40.1 ; \mathrm { A } = 88.2 ^ { \circ } , \mathrm { C } = 126.8 ^ { \circ } , \mathrm { a } = 40.1A=91.8∘,C=53.2∘,a=40.1;A=88.2∘,C=126.8∘,a=40.1 
C)    A=108.2∘,C=36.8∘,a=13.9;A=1.8∘,C=143.2∘,a=13.9\mathrm { A } = 108.2 ^ { \circ } , \mathrm { C } = 36.8 ^ { \circ } , \mathrm { a } = 13.9 ; \mathrm { A } = 1.8 ^ { \circ } , \mathrm { C } = 143.2 ^ { \circ } , \mathrm { a } = 13.9A=108.2∘,C=36.8∘,a=13.9;A=1.8∘,C=143.2∘,a=13.9 
D)    A=108.2∘,C=36.8∘,a=38.1;A=1.8∘,C=143.2∘,a=1.3\mathrm { A } = 108.2 ^ { \circ } , \mathrm { C } = 36.8 ^ { \circ } , \mathrm { a } = 38.1 ; \mathrm { A } = 1.8 ^ { \circ } , \mathrm { C } = 143.2 ^ { \circ } , \mathrm { a } = 1.3A=108.2∘,C=36.8∘,a=38.1;A=1.8∘,C=143.2∘,a=1.3E) All of the above
F) A) and B)

Question 2

Prove the identity.
-$$\frac { \csc x - 1 } { \csc x + 1 } = \frac { \cot ^ { 2 } x } { \csc ^ { 2 } x + 2 \csc x + 1 }$$

Accepted Answer

The answer of Prove the identity.
-\[\frac { \csc x -...

Question 3

Find the area. Round your answer to the nearest hundredth if necessary.
-Find the area of the triangle with the following measurements:  A=50∘,b=25ft,c=15ft\mathrm { A } = 50 ^ { \circ } , \mathrm { b } = 25 \mathrm { ft } , \mathrm { c } = 15 \mathrm { ft }A=50∘,b=25ft,c=15ft

Accepted Answer

A)    120.52ft2120.52 \mathrm { ft } ^ { 2 }120.52ft2 
B)    375ft2375 \mathrm { ft } ^ { 2 }375ft2 
C)    287.27ft2287.27 \mathrm { ft } ^ { 2 }287.27ft2 
D)    143.63ft2143.63 \mathrm { ft } ^ { 2 }143.63ft2E) A) and B)
F) A) and C)

Question 4

Prove the identity.
-$$\cos 4 x = 1 - 2 \sin ^ { 2 } 2 x$$

Accepted Answer

The answer of Prove the identity.
-\[\cos 4 x = 1...

Question 5

Provide an appropriate response.
-$$\text { Show that } \csc ( A + B ) = \frac { \csc A \csc B } { \cot B + \cot A } \text {. }$$

Accepted Answer

To show that \csc(A + B) = \frac{\csc A ...

Question 6

Write the expression as the sine, cosine, or tangent of an angle.
- cos⁡π7cos⁡π11+sin⁡π7sin⁡π11\cos \frac { \pi } { 7 } \cos \frac { \pi } { 11 } + \sin \frac { \pi } { 7 } \sin \frac { \pi } { 11 }cos7π​cos11π​+sin7π​sin11π​

Accepted Answer

A)    sin⁡18π77\sin \frac { 18 \pi } { 77 }sin7718π​ 
B)    cos⁡4π77\cos \frac { 4 \pi } { 77 }cos774π​ 
C)    sin⁡4π77\sin \frac { 4 \pi } { 77 }sin774π​ 
D)    cos⁡18π77\cos \frac { 18 \pi } { 77 }cos7718π​E) B) and D)
F) B) and C)

Question 7

State whether the given measurements determine zero, one, or two triangles.
- B=87∘,b=27,c=28B = 87 ^ { \circ } , b = 27 , c = 28B=87∘,b=27,c=28

Accepted Answer

A)   Zero
B)   Two
C)   One D) A) and B)
E) A) and C)

Question 8

Prove the identity.
-$$4 \csc 2 x = 2 \csc ^ { 2 } x \tan x$$

Accepted Answer

The answer of Prove the identity.
-\[4 \csc 2 x =...

Question 9

Find all solutions to the equation.
- sin⁡x=32\sin x = \frac { \sqrt { 3 } } { 2 } \quadsinx=23​​  (Express your answer in radians, in exact form.)

Accepted Answer

A)    {π3+2nπ,5π3+2nπ∣n=0,±1,±2,…}\left\{ \frac { \pi } { 3 } + 2 \mathrm { n } \pi , \frac { 5 \pi } { 3 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}{3π​+2nπ,35π​+2nπ∣n=0,±1,±2,…} 
B)    {π3+nπn=0,±1,±2,…}\left\{ \frac { \pi } { 3 } + n \pi \quad \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}{3π​+nπn=0,±1,±2,…} 
C)    {π6+2nπ,5π6+2nπ∣n=0,±1,±2,…}\left\{ \frac { \pi } { 6 } + 2 \mathrm { n } \pi , \frac { 5 \pi } { 6 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}{6π​+2nπ,65π​+2nπ∣n=0,±1,±2,…} 
D)    {π3+2nπ,2π3+2nπ∣n=0,±1,±2,…}\left\{ \frac { \pi } { 3 } + 2 \mathrm { n } \pi , \frac { 2 \pi } { 3 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}{3π​+2nπ,32π​+2nπ∣n=0,±1,±2,…}E) C) and D)
F) A) and B)

Question 10

The given measurements may or may not determine a triangle. If not, then state that no triangle is formed. If a triangle is
formed, then use the Law of Sines to solve the triangle, if it is possible, or state that the Law of Sines cannot be used.
-

Accepted Answer

A)    B=57∘,a≈5.4\mathrm { B } = 57 ^ { \circ } , \mathrm { a } \approx 5.4B=57∘,a≈5.4 
B)    B=57∘,a≈53.1\mathrm { B } = 57 ^ { \circ } , \mathrm { a } \approx 53.1B=57∘,a≈53.1 
C)    B=33∘,a≈53.1B = 33 ^ { \circ } , a \approx 53.1B=33∘,a≈53.1 
D)   No triangle is formed. E) All of the above
F) A) and B)

Question 11

Solve the problem.
-A building has a ramp to its front doors to accommodate the handicapped. If the distance from the building to the end of the ramp is 13 feet and the height from the ground to the front doors is 4 feet, how long is the ramp?
(Round to the nearest tenth.)

Accepted Answer

A)   13.6 ft
B)   12.4 ft
C)   4.1 ft
D)   5.7 ftE) A) and B)
F) C) and D)

Question 12

Find an exact value.
- cos⁡165∘\cos 165 ^ { \circ }cos165∘

Accepted Answer

A)    −6−24\frac { - \sqrt { 6 } - \sqrt { 2 } } { 4 }4−6c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​−2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​ 
B)    2+64\frac { \sqrt { 2 } + \sqrt { 6 } } { 4 }42c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​+6c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​ 
C)    6−24\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }46c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​−2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​ 
D)    2−64\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }42c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​−6c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​E) B) and C)
F) A) and D)

Question 13

Find all solutions in the interval [0, 2π]
- sin⁡2x+sin⁡x=0\sin ^ { 2 } x + \sin x = 0sin2x+sinx=0

Accepted Answer

A)    x=0,π,3π2x = 0 , \pi , \frac { 3 \pi } { 2 }x=0,π,23π​ 
B)    x=0,π,π3,5π3x = 0 , \pi , \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }x=0,π,3π​,35π​ 
C)    x=0,π,4π3,5π3x = 0 , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }x=0,π,34π​,35π​ 
D)    x=0,π,π3,2π3x = 0 , \pi , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }x=0,π,3π​,32π​E) A) and B)
F) A) and C)

Question 14

Find all solutions to the equation.
- cos⁡x=sin⁡x\cos x = \sin xcosx=sinx

Accepted Answer

A)    {π4+2nπ,7π4+2nπ∣n=0,±1,±2,…}\left\{ \frac { \pi } { 4 } + 2 \mathrm { n } \pi , \frac { 7 \pi } { 4 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}{4π​+2nπ,47π​+2nπ∣n=0,±1,±2,…} 
B)    {π4+nπ∣n=0,±1,±2,…}\left\{ \frac { \pi } { 4 } + n \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}{4π​+nπ∣n=0,±1,±2,…} 
C)    {π2+nπ∣n=0,±1,±2,….}\left\{ \frac { \pi } { 2 } + \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \right\}{2π​+nπ∣n=0,±1,±2,….} 
D)    {3π4+2nπ,5π4+2nπ∣n=0,±1,±2,….}\left\{ \frac { 3 \pi } { 4 } + 2 \mathrm { n } \pi , \frac { 5 \pi } { 4 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \right\}{43π​+2nπ,45π​+2nπ∣n=0,±1,±2,….}E) C) and D)
F) None of the above

Question 15

Find all solutions in the interval [0, 2π) .
- cot⁡(x2) =1−cos⁡x1+cos⁡x\cot \left( \frac { x } { 2 } \right)  = \frac { 1 - \cos x } { 1 + \cos x }cot(2x​) =1+cosx1−cosx​

Accepted Answer

A)    π4,5π4\frac { \pi } { 4 } , \frac { 5 \pi } { 4 }4π​,45π​ 
B)   0
C)    π2\frac { \pi } { 2 }2π​ 
D)    3π4,7π4\frac { 3 \pi } { 4 } , \frac { 7 \pi } { 4 }43π​,47π​E) B) and D)
F) B) and C)

Question 16

Simplify the expression.
- cot⁡xsec⁡2x+cot⁡xcsc⁡2x\frac { \cot x } { \sec ^ { 2 } x } + \frac { \cot x } { \csc ^ { 2 } x }sec2xcotx​+csc2xcotx​

Accepted Answer

A)    cot⁡x\cot xcotx 
B)    tan⁡x	an xtanx 
C)    sin⁡x\sin xsinx 
D)    cot⁡2x\cot ^ { 2 } xcot2xE) A) and C)
F) A) and B)

Question 17

Given the SSS parts of a triangle, is it better to use the law of sines or the law of cosines as the first step in
solving the triangle? Explain.

Accepted Answer

Use the law of cosin...

Question 18

Write each expression in factored form as an algebraic expression of a single trigonometric function.
- csc⁡2x−1\csc ^ { 2 } x - 1csc2x−1

Accepted Answer

A)    (csc⁡x+1) (csc⁡x−1) ( \csc x + 1 )  ( \csc x - 1 ) (cscx+1) (cscx−1)  
B)    cot⁡x\cot xcotx 
C)    (cot⁡x+1) (cot⁡x−1) ( \cot x + 1 )  ( \cot x - 1 ) (cotx+1) (cotx−1)  
D)    csc⁡x−1\csc x - 1cscx−1E) B) and C)
F) None of the above

Question 19

Find all solutions to the equation in the interval [0, 2) .
- sin⁡2x+cos⁡3x=0\sin 2 x + \cos 3 x = 0sin2x+cos3x=0

Accepted Answer

A)    0.3π,0.5π,0.7π,1.9π0.3 \pi , 0.5 \pi , 0.7 \pi , 1.9 \pi0.3π,0.5π,0.7π,1.9π 
B)    0.3π,0.5π,0.7π,1.1π,1.5π,1.9π0.3 \pi , 0.5 \pi , 0.7 \pi , 1.1 \pi , 1.5 \pi , 1.9 \pi0.3π,0.5π,0.7π,1.1π,1.5π,1.9π 
C)    0.3π,0.7π,1.1π,1.9π0.3 \pi , 0.7 \pi , 1.1 \pi , 1.9 \pi0.3π,0.7π,1.1π,1.9π 
D)    0.5π,1.5π0.5 \pi , 1.5 \pi0.5π,1.5πE) A) and B)
F) None of the above

Question 20

Find an exact value.
- sin⁡105∘\sin 105 ^ { \circ }sin105∘

Accepted Answer

A)    −6−24\frac { - \sqrt { 6 } - \sqrt { 2 } } { 4 }4−6c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​−2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​ 
B)    6−24\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }46c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​−2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​ 
C)    −6+24\frac { - \sqrt { 6 } + \sqrt { 2 } } { 4 }4−6c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​+2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​ 
D)    6+24\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }46c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​+2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​E) A) and B)
F) A) and C)

Exam 5: Analytic Trigonometry

Find all solutions in the interval [0, 2π] - $\sin ^ { 2 } x + \sin x = 0$

Correct Answer
verified

Given the SSS parts of a triangle, is it better to use the law of sines or the law of cosines as the first step in solving the triangle? Explain.

Correct Answer
verified
Use the law of cosin...
View Answer

Solve the problem. -A building has a ramp to its front doors to accommodate the handicapped. If the distance from the building to the end of the ramp is 13 feet and the height from the ground to the front doors is 4 feet, how long is the ramp? (Round to the nearest tenth.)

Correct Answer
verified

Prove the identity. - $\frac { \csc x - 1 } { \csc x + 1 } = \frac { \cot ^ { 2 } x } { \csc ^ { 2 } x + 2 \csc x + 1 }$

Correct Answer
verified

Simplify the expression. - $\frac { \cot x } { \sec ^ { 2 } x } + \frac { \cot x } { \csc ^ { 2 } x }$

Correct Answer
verified

Find all solutions to the equation. - $\sin x = \frac { \sqrt { 3 } } { 2 } \quad$ (Express your answer in radians, in exact form.)

Correct Answer
verified

Find all solutions to the equation. - $\cos x = \sin x$

Correct Answer
verified

Find all solutions in the interval [0, 2π) . - $\cot \left( \frac { x } { 2 } \right) = \frac { 1 - \cos x } { 1 + \cos x }$

Correct Answer
verified

Two triangles can be formed using the given measurements. Solve both triangles. - $\mathrm { B } = 35 ^ { \circ } , \mathrm { b } = 23 , \mathrm { c } = 24$

Correct Answer
verified

Find the area. Round your answer to the nearest hundredth if necessary. -Find the area of the triangle with the following measurements: $\mathrm { A } = 50 ^ { \circ } , \mathrm { b } = 25 \mathrm { ft } , \mathrm { c } = 15 \mathrm { ft }$

Correct Answer
verified

Write each expression in factored form as an algebraic expression of a single trigonometric function. - $\csc ^ { 2 } x - 1$

Correct Answer
verified

Find an exact value. - $\sin 105 ^ { \circ }$

Correct Answer
verified

State whether the given measurements determine zero, one, or two triangles. - $B = 87 ^ { \circ } , b = 27 , c = 28$

Correct Answer
verified

Find all solutions to the equation in the interval [0, 2) . - $\sin 2 x + \cos 3 x = 0$

Correct Answer
verified

Find an exact value. - $\cos 165 ^ { \circ }$

Correct Answer
verified

Prove the identity. - $\cos 4 x = 1 - 2 \sin ^ { 2 } 2 x$

Correct Answer
verified

Prove the identity. - $4 \csc 2 x = 2 \csc ^ { 2 } x \tan x$

Correct Answer
verified

Provide an appropriate response. - $\text { Show that } \csc ( A + B ) = \frac { \csc A \csc B } { \cot B + \cot A } \text {. }$

Correct Answer
Answered by ExamLex AI
To show that \csc(A + B) = \frac{\csc A ...
View Answer

Correct Answer
verified

Write the expression as the sine, cosine, or tangent of an angle. - $\cos \frac { \pi } { 7 } \cos \frac { \pi } { 11 } + \sin \frac { \pi } { 7 } \sin \frac { \pi } { 11 }$

Correct Answer
verified

Exam 5: Analytic Trigonometry

Find all solutions in the interval [0, 2π] - sin⁡2x+sin⁡x=0\sin ^ { 2 } x + \sin x = 0sin2x+sinx=0

Correct AnswerverifiedShow Answer

Given the SSS parts of a triangle, is it better to use the law of sines or the law of cosines as the first step in solving the triangle? Explain.

Correct AnswerverifiedUse the law of cosin...View AnswerShow Answer

Solve the problem. -A building has a ramp to its front doors to accommodate the handicapped. If the distance from the building to the end of the ramp is 13 feet and the height from the ground to the front doors is 4 feet, how long is the ramp? (Round to the nearest tenth.)

Correct AnswerverifiedShow Answer

Prove the identity. - csc⁡x−1csc⁡x+1=cot⁡2xcsc⁡2x+2csc⁡x+1\frac { \csc x - 1 } { \csc x + 1 } = \frac { \cot ^ { 2 } x } { \csc ^ { 2 } x + 2 \csc x + 1 }cscx+1cscx−1​=csc2x+2cscx+1cot2x​

Correct AnswerverifiedShow Answer

Simplify the expression. - cot⁡xsec⁡2x+cot⁡xcsc⁡2x\frac { \cot x } { \sec ^ { 2 } x } + \frac { \cot x } { \csc ^ { 2 } x }sec2xcotx​+csc2xcotx​

Correct AnswerverifiedShow Answer

Find all solutions to the equation. - sin⁡x=32\sin x = \frac { \sqrt { 3 } } { 2 } \quadsinx=23​​ (Express your answer in radians, in exact form.)

Correct AnswerverifiedShow Answer

Find all solutions to the equation. - cos⁡x=sin⁡x\cos x = \sin xcosx=sinx

Correct AnswerverifiedShow Answer

Find all solutions in the interval [0, 2π) . - cot⁡(x2) =1−cos⁡x1+cos⁡x\cot \left( \frac { x } { 2 } \right) = \frac { 1 - \cos x } { 1 + \cos x }cot(2x​) =1+cosx1−cosx​

Correct AnswerverifiedShow Answer

Two triangles can be formed using the given measurements. Solve both triangles. - B=35∘,b=23,c=24\mathrm { B } = 35 ^ { \circ } , \mathrm { b } = 23 , \mathrm { c } = 24B=35∘,b=23,c=24

Correct AnswerverifiedShow Answer

Find the area. Round your answer to the nearest hundredth if necessary. -Find the area of the triangle with the following measurements: A=50∘,b=25ft,c=15ft\mathrm { A } = 50 ^ { \circ } , \mathrm { b } = 25 \mathrm { ft } , \mathrm { c } = 15 \mathrm { ft }A=50∘,b=25ft,c=15ft

Correct AnswerverifiedShow Answer

Write each expression in factored form as an algebraic expression of a single trigonometric function. - csc⁡2x−1\csc ^ { 2 } x - 1csc2x−1

Correct AnswerverifiedShow Answer

Find an exact value. - sin⁡105∘\sin 105 ^ { \circ }sin105∘

Correct AnswerverifiedShow Answer

State whether the given measurements determine zero, one, or two triangles. - B=87∘,b=27,c=28B = 87 ^ { \circ } , b = 27 , c = 28B=87∘,b=27,c=28

Correct AnswerverifiedShow Answer

Find all solutions to the equation in the interval [0, 2) . - sin⁡2x+cos⁡3x=0\sin 2 x + \cos 3 x = 0sin2x+cos3x=0

Correct AnswerverifiedShow Answer

Find an exact value. - cos⁡165∘\cos 165 ^ { \circ }cos165∘

Correct AnswerverifiedShow Answer

Prove the identity. - cos⁡4x=1−2sin⁡22x\cos 4 x = 1 - 2 \sin ^ { 2 } 2 xcos4x=1−2sin22x

Correct AnswerverifiedShow Answer

Prove the identity. - 4csc⁡2x=2csc⁡2xtan⁡x4 \csc 2 x = 2 \csc ^ { 2 } x \tan x4csc2x=2csc2xtanx

Correct AnswerverifiedShow Answer

Provide an appropriate response. - Show that csc⁡(A+B)=csc⁡Acsc⁡Bcot⁡B+cot⁡A. \text { Show that } \csc ( A + B ) = \frac { \csc A \csc B } { \cot B + \cot A } \text {. } Show that csc(A+B)=cotB+cotAcscAcscB​.

Correct AnswerAnswered by ExamLex AITo show that \csc(A + B) = \frac{\csc A ...View AnswerShow Answer

The given measurements may or may not determine a triangle. If not, then state that no triangle is formed. If a triangle is formed, then use the Law of Sines to solve the triangle, if it is possible, or state that the Law of Sines cannot be used. -

Correct AnswerverifiedShow Answer

Write the expression as the sine, cosine, or tangent of an angle. - cos⁡π7cos⁡π11+sin⁡π7sin⁡π11\cos \frac { \pi } { 7 } \cos \frac { \pi } { 11 } + \sin \frac { \pi } { 7 } \sin \frac { \pi } { 11 }cos7π​cos11π​+sin7π​sin11π​

Correct AnswerverifiedShow Answer

Find all solutions in the interval [0, 2π] - $\sin ^ { 2 } x + \sin x = 0$

Correct Answer
verified

Correct Answer
verified
Use the law of cosin...
View Answer

Correct Answer
verified

Prove the identity. - $\frac { \csc x - 1 } { \csc x + 1 } = \frac { \cot ^ { 2 } x } { \csc ^ { 2 } x + 2 \csc x + 1 }$

Correct Answer
verified

Simplify the expression. - $\frac { \cot x } { \sec ^ { 2 } x } + \frac { \cot x } { \csc ^ { 2 } x }$

Correct Answer
verified

Find all solutions to the equation. - $\sin x = \frac { \sqrt { 3 } } { 2 } \quad$ (Express your answer in radians, in exact form.)

Correct Answer
verified

Find all solutions to the equation. - $\cos x = \sin x$

Correct Answer
verified

Find all solutions in the interval [0, 2π) . - $\cot \left( \frac { x } { 2 } \right) = \frac { 1 - \cos x } { 1 + \cos x }$

Correct Answer
verified

Two triangles can be formed using the given measurements. Solve both triangles. - $\mathrm { B } = 35 ^ { \circ } , \mathrm { b } = 23 , \mathrm { c } = 24$

Correct Answer
verified

Find the area. Round your answer to the nearest hundredth if necessary. -Find the area of the triangle with the following measurements: $\mathrm { A } = 50 ^ { \circ } , \mathrm { b } = 25 \mathrm { ft } , \mathrm { c } = 15 \mathrm { ft }$

Correct Answer
verified

Write each expression in factored form as an algebraic expression of a single trigonometric function. - $\csc ^ { 2 } x - 1$

Correct Answer
verified

Find an exact value. - $\sin 105 ^ { \circ }$

Correct Answer
verified

State whether the given measurements determine zero, one, or two triangles. - $B = 87 ^ { \circ } , b = 27 , c = 28$

Correct Answer
verified

Find all solutions to the equation in the interval [0, 2) . - $\sin 2 x + \cos 3 x = 0$

Correct Answer
verified

Find an exact value. - $\cos 165 ^ { \circ }$

Correct Answer
verified

Prove the identity. - $\cos 4 x = 1 - 2 \sin ^ { 2 } 2 x$

Correct Answer
verified

Prove the identity. - $4 \csc 2 x = 2 \csc ^ { 2 } x \tan x$

Correct Answer
verified

Provide an appropriate response. - $\text { Show that } \csc ( A + B ) = \frac { \csc A \csc B } { \cot B + \cot A } \text {. }$

Correct Answer
Answered by ExamLex AI
To show that \csc(A + B) = \frac{\csc A ...
View Answer

Correct Answer
verified

Write the expression as the sine, cosine, or tangent of an angle. - $\cos \frac { \pi } { 7 } \cos \frac { \pi } { 11 } + \sin \frac { \pi } { 7 } \sin \frac { \pi } { 11 }$

Correct Answer
verified