In Exercises 17–20, θ is an acute angle and sin θ and cos θ are given. Use identities to find tan θ, csc θ, sec θ, and cot θ. Where necessary, rationalize denominators.sin θ = 3/5, cos θ = 4/5

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1. Recall the identity for tangent: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute the given values: \( \tan \theta = \frac{3/5}{4/5} \).
2. Simplify the expression for \( \tan \theta \) by dividing the fractions: \( \tan \theta = \frac{3}{4} \).
3. Use the identity for cosecant: \( \csc \theta = \frac{1}{\sin \theta} \). Substitute the given value: \( \csc \theta = \frac{1}{3/5} \).
4. Simplify the expression for \( \csc \theta \) by taking the reciprocal: \( \csc \theta = \frac{5}{3} \).
5. Use the identity for secant: \( \sec \theta = \frac{1}{\cos \theta} \). Substitute the given value: \( \sec \theta = \frac{1}{4/5} \), and simplify to find \( \sec \theta = \frac{5}{4} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, which states that sin²θ + cos²θ = 1, and the definitions of the tangent, cosecant, secant, and cotangent functions in terms of sine and cosine. These identities are essential for deriving other trigonometric values from given ones.

Definitions of Trigonometric Functions

The primary trigonometric functions are defined based on a right triangle or the unit circle. For an acute angle θ, the sine (sin θ) is the ratio of the opposite side to the hypotenuse, while the cosine (cos θ) is the ratio of the adjacent side to the hypotenuse. The tangent (tan θ) is defined as the ratio of sine to cosine, and the cosecant (csc θ), secant (sec θ), and cotangent (cot θ) are the reciprocals of sine, cosine, and tangent, respectively.

Rationalizing Denominators

Rationalizing the denominator is a mathematical technique used to eliminate any radical expressions from the denominator of a fraction. This is often done by multiplying the numerator and denominator by a suitable form of 1, such as the conjugate of the denominator. In trigonometry, this process is important for simplifying expressions involving trigonometric functions, ensuring that the final answers are presented in a standard form.