Q: Which of the following is the correct difference formula for the tangent function?

$$\tan(\alpha - \beta) = \dfrac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$$

Q: Solve for $$\theta in $$the equation $$\sin \theta + 3 \cos \theta = 1$$ for $$0 \leq \theta \u003c 2\pi.$$

$$\theta = \dfrac{\pi}{2}, \dfrac{11\pi}{6}$$

Question 1

Which of the following is the correct sum formula for the cosine function?

Accepted Answer

$$\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$$

Question 2

Find the exact value of $$\sin 75^\circ$$ using a sum or difference formula.

Accepted Answer

$$\dfrac{\sqrt{6} + \sqrt{2}}{4}$$

Question 3

Which of the following is the correct difference formula for the tangent function?

Accepted Answer

$$	an(\alpha - \beta) = \dfrac{	an \alpha - 	an \beta}{1 + 	an \alpha 	an \beta}$$

Question 4

Evaluate $$\cos 15^\circ$$ using the appropriate sum or difference formula.

Accepted Answer

$$\dfrac{\sqrt{6} - \sqrt{2}}{4}$$

Question 5

Which of the following is the correct sum formula for the sine function?

Accepted Answer

$$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$$

Question 6

Find the exact value of $$	an 165^\circ$$ using a sum or difference formula.

Accepted Answer

$$-\dfrac{\sqrt{3} + 1}{\sqrt{3}}$$

Question 7

Simplify $$\cos 35^\circ \cos 55^\circ - \sin 35^\circ \sin 55^\circ$$ using a sum or difference formula.

Accepted Answer

$$\cos 90^\circ = 0$$

Question 8

Given $$\sin \alpha = \dfrac{5}{13}$$ and $$\cos \alpha = \dfrac{12}{13}$$ with $$\alpha in $$Quadrant I, and $$\sin \beta = \dfrac{12}{13}$$ and $$\cos \beta = \dfrac{5}{13}$$ with $$\beta in $$Quadrant I, find $$\sin \alpha \cos \beta - \cos \alpha \sin \beta.$$

Accepted Answer

$$-\dfrac{119}{169}$$

Question 9

Which real-world scenario could best be modeled using the sum and difference formulas for trigonometric functions?

Accepted Answer

Calculating the angle of elevation from two different points to the top of a building.

Question 10

Establish the identity: $$\cos \alpha \cot \beta + \sin \alpha = \dfrac{\cos(\alpha + \beta)}{\sin \beta}.$$

Accepted Answer

True, by expressing $$\cot \beta as$$ $$\dfrac{\cos \beta}{\sin \beta}$$ and simplifying.

Question 11

Solve for $$	heta in $$the equation $$\sin 	heta + 3 \cos 	heta = 1$$ for $$0 \leq 	heta \u003c 2\pi.$$

Accepted Answer

$$	heta = \dfrac{\pi}{2}, \dfrac{11\pi}{6}$$

Question 12

If $$	an \alpha = 2$$ and $$	an \beta = 3$$, find $$	an(\alpha + \beta).$$

Accepted Answer

$$-1$$

Question 13

Which of the following is the correct difference formula for the sine function?

Accepted Answer

$$\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$$

Question 14

Evaluate $$	an(75^\circ)$$ using the sum formula for tangent.

Accepted Answer

$$2 + \sqrt{3}$$

Question 15

A satellite is tracking two ships. The angle between the lines of sight to the ships is $$40^\circ. If $$the satellite measures the angle of elevation to one ship as $$30^\circ$$ and to the other as $$70^\circ$$, which formula would you use to find the sine of the angle between the ships as seen from the satellite?

Accepted Answer

$$\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$$

Sum and Difference Formulas in Trigonometry

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