Q: Which of the following is the correct half-angle identity for $$\tan\left(\dfrac{x}{2}\right)$$?

$$\tan\left(\dfrac{x}{2}\right) = \dfrac{1 - \cos(x)}{1 + \cos(x)}$$

Q: Given $$\tan(22.5^\circ)$$, which identity can be used to find its exact value?

Half-angle identity: $$\tan\left(\dfrac{x}{2}\right) = \dfrac{1 - \cos(x)}{\sin(x)}$$

Q: Which of the following is the correct expression for $$\tan(A+B)$$?

$$\tan(A+B) = \dfrac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}$$

Question 1

Which of the following is the correct double-angle identity for $$\sin(2x)$$?

Accepted Answer

$$\sin(2x) = 2\sin(x)\cos(x)$$

Question 2

Given $$	an(x) = \dfrac{3}{4}$$ and $$x is in $$Quadrant I, what is $$\sin(x)$$?

Accepted Answer

$$\dfrac{3}{5}$$

Question 3

Which of the following is the correct formula for $$\cos(A+B)$$?

Accepted Answer

$$\cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B)$$

Question 4

If $$\sin(x) = \dfrac{5}{13}$$ and $$x is in $$Quadrant II, what is $$\cos(x)$$?

Accepted Answer

$$-\dfrac{12}{13}$$

Question 5

Simplify $$\dfrac{\sin(75^\circ)}{\cos(15^\circ)}$$ using sum or difference identities.

Accepted Answer

$$1$$

Question 6

Which of the following is the correct half-angle identity for $$	an\left(\dfrac{x}{2}ight)$$?

Accepted Answer

$$	an\left(\dfrac{x}{2}ight) = \dfrac{1 - \cos(x)}{1 + \cos(x)}$$

Question 7

A ramp rises at an angle of $$30^\circ to $$the ground. If the ramp is 10 meters long, what is the vertical height?

Accepted Answer

$$5$$ meters

Question 8

If $$\sin(	heta) = 0.6$$ and $$	heta is in $$Quadrant II, what is $$	an(	heta)$$?

Accepted Answer

$$-0.75$$

Question 9

Which of the following is the correct expression for $$\sin(A+B)$$?

Accepted Answer

$$\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B)$$

Question 10

Given $$	an(22.5^\circ)$$, which identity can be used to find its exact value?

Accepted Answer

Half-angle identity: $$	an\left(\dfrac{x}{2}ight) = \dfrac{1 - \cos(x)}{\sin(x)}$$

Question 11

If $$\cos(x) = -0.8$$ and $$x is in $$Quadrant III, what is $$\sin(x)$$?

Accepted Answer

$$-0.6$$

Question 12

A surveyor measures the angle of elevation to the top of a building as $$40^\circ$$ from a point 50 meters away. What is the height of the building?

Accepted Answer

$$50 	an(40^\circ)$$ meters

Question 13

Which of the following is the correct expression for $$\cos(2x)$$ using the double-angle identity?

Accepted Answer

$$\cos(2x) = 2\cos^2$(x) - 1$$

Question 14

If $$\sin(x) = \dfrac{1}{2}$$ and $$x is in $$Quadrant I, what is $$x in $$degrees?

Accepted Answer

$$30^\circ$$

Question 15

Given $$	an(x) = -2$$ and $$x is in $$Quadrant II, what is $$\sin(x)$$?

Accepted Answer

$$\dfrac{2}{\sqrt{5}}$$

Question 16

Which of the following is the correct sum-to-product identity for $$\sin(A) + \sin(B)$$?

Accepted Answer

$$\sin(A) + \sin(B) = 2\sin\left(\dfrac{A+B}{2}\right)\cos\left(\dfrac{A-B}{2}\right)$$

Question 17

A ship sails 100 km east, then turns $$60^\circ$$ north of east and sails another 100 km. What is the straight-line distance from the starting point to the ship's final position?

Accepted Answer

$$100\sqrt{3} km$$

Question 18

If $$\sin(2x) = 0.8$$, what is $$\sin(x)$$?

Accepted Answer

$$0.4 or$$ $$-0.4$$

Question 19

Which of the following is the correct expression for $$	an(A+B)$$?

Accepted Answer

$$	an(A+B) = \dfrac{	an(A) + 	an(B)}{1 - 	an(A)	an(B)}$$

Trigonometry Exam 3 Study Guide: Identities, Equations, and Applications

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