Question

In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c. cos t + cos(t + 1000𝜋) - tan t - tan(t + 999𝜋) - sin t + 4 sin(t - 1000𝜋)

Accepted Answer

Recall the periodic properties of sine, cosine, and tangent functions: sine and cosine have a period of $$2\pi$$, and tangent has a period of $$\pi. $$This means $$\sin(t + 2k\pi) = \sin t$$, $$\cos(t + 2k\pi) = \cos t$$, and $$	an(t + k\pi) = 	an t$$ for any integer $$k. $$Simplify each trigonometric term with the given large multiples of $$\pi$$ using the periodicity: 
- $$\cos(t + 1000\pi)$$ can be rewritten using the fact that $$1000\pi = 500 	imes 2\pi$$, so $$\cos(t + 1000\pi) = \cos t$$.
- $$	an(t + 999\pi)$$ can be rewritten using $$999\pi = 999 	imes \pi$$, so $$	an(t + 999\pi) = 	an t$$.
- $$\sin(t - 1000\pi)$$ can be rewritten as $$\sin(t - 1000\pi) = \sin t$$ because $$-1000\pi = -500 	imes 2\pi. $$Substitute the simplified expressions back into the original expression:
$$\cos t + \cos t - 	an t - 	an t - \sin t + 4 \sin t. $$Group like terms together:
Combine the cosine terms: $$\cos t + \cos t = 2b ($$since $$\cos t = b).
$$Combine the tangent terms: $$-	an t - 	an t = -2c ($$since $$	an t = c).
$$Combine the sine terms: $$-\sin t + 4 \sin t = 3a ($$since $$\sin t = a). $$Write the final simplified expression in terms of $$a$$, $$b$$, and $$c$$:
$$2b - 2c + 3a.$$

In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c.
cos t + cos(t + 1000𝜋) - tan t - tan(t + 999𝜋) - sin t + 4 sin(t - 1000𝜋)

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주요 개념

Periodic Properties of Trigonometric Functions

Trigonometric Function Definitions and Relationships

Angle Addition and Subtraction Formulas