Use the function value(s) and the trigonometric identities...

Use the function value(s) and the trigonometric identities to evaluate each trigonometric function.
$\sec \theta=5, \tan \theta=2 \sqrt{6}$
(a) $\cos \theta$
(b) $\cot \theta$
(c) $\cot \left(90^{\circ}-\theta\right)$
(d) $\sin \theta$

Key Concepts

Reciprocal Identities

Reciprocal identities relate a trigonometric function to its reciprocal function. For example, the cosine of an angle is the reciprocal of the secant (cos? = 1/sec?), and similarly, the sine and cosecant, and the tangent and cotangent are paired through reciprocal relationships. These identities allow the conversion between functions which can simplify the process of solving problems where only one function’s value is given.

Quotient Identities

Quotient identities express the tangent and cotangent functions in terms of sine and cosine. Specifically, tan? is defined as sin? divided by cos?, while cot? is the reciprocal (or cos? divided by sin?). These identities are fundamental for converting between ratios of sides in a right triangle and are frequently used in problems involving relationships between different trigonometric functions.

Cofunction Identities

Cofunction identities state that the trigonometric function of an angle is equal to the cofunction of its complement. For instance, the sine of an angle is equal to the cosine of its complement, and similarly, cotangent of an angle is the tangent of the complement (cot(90° - ?) = tan?). These identities are useful when dealing with complementary angles and can simplify expressions involving trigonometric functions of such angles.

Pythagorean Identities

Pythagorean identities derive from the Pythagorean theorem applied to the sides of a right triangle, leading to key relationships such as sin²? + cos²? = 1. They provide a way to relate sine, cosine, tangent, and their reciprocals, which is essential for finding unknown function values from given information and for verifying identities in trigonometry.

Trigonometric Function Definitions

Trigonometric functions define the ratios of the sides of a right triangle in relation to its angles, or in the unit circle they are represented as coordinates. Understanding these definitions is crucial since they form the basis for all trigonometric identities and relationships. This conceptual framework helps in evaluating and transforming trigonometric expressions given certain known values.

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