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)

Verified step by step guidance

1. Recognize that trigonometric functions are periodic. The cosine and sine functions have a period of $2\pi$, and the tangent function has a period of $\pi$. This means $\cos(t + 1000\pi) = \cos t$, $\tan(t + 999\pi) = \tan t$, and $\sin(t - 1000\pi) = \sin t$.
2. Substitute the periodic identities into the expression: $\cos t + \cos t - \tan t - \tan t - \sin t + 4\sin t$.
3. Combine like terms: $2\cos t - 2\tan t + 3\sin t$.
4. Substitute the given values: $\cos t = b$, $\tan t = c$, and $\sin t = a$.
5. Rewrite the expression in terms of $a$, $b$, and $c$: $2b - 2c + 3a$.

Key Concepts

Periodic Functions

Trigonometric functions such as sine, cosine, and tangent are periodic, meaning they repeat their values in regular intervals. For example, sin(t + 2π) = sin(t) and cos(t + 2π) = cos(t). This property is crucial for simplifying expressions involving angles that differ by multiples of π or 2π, as it allows us to reduce the angles to their equivalent values within one period.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables. Key identities include the Pythagorean identity (sin²t + cos²t = 1) and angle addition formulas. These identities are essential for rewriting expressions in terms of sine, cosine, and tangent, facilitating simplification and solving trigonometric equations.

Angle Transformation

Angle transformation involves manipulating angles using properties of trigonometric functions. For instance, the transformations sin(t - 1000π) and cos(t + 1000π) can be simplified using periodicity. Understanding how to transform angles helps in rewriting complex expressions into simpler forms, making it easier to express them in terms of a, b, and c.