In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. sec θ = -3, tan θ > 0

Verified step by step guidance

1. <insert step 1: Understand the given information. We know that \( \sec \theta = -3 \) and \( \tan \theta > 0 \). The secant function is the reciprocal of the cosine function, so \( \cos \theta = -\frac{1}{3} \).>
2. <insert step 2: Determine the quadrant where \( \theta \) is located. Since \( \sec \theta = -3 \) (meaning \( \cos \theta < 0 \)) and \( \tan \theta > 0 \), \( \theta \) must be in the third quadrant, where both sine and cosine are negative, but tangent is positive.>
3. <insert step 3: Use the Pythagorean identity to find \( \sin \theta \). The identity is \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \cos \theta = -\frac{1}{3} \) into the equation: \( \sin^2 \theta + \left(-\frac{1}{3}\right)^2 = 1 \).>
4. <insert step 4: Solve for \( \sin \theta \). Simplify the equation from step 3 to find \( \sin^2 \theta \), and then take the square root to find \( \sin \theta \). Remember that in the third quadrant, \( \sin \theta \) is negative.>
5. <insert step 5: Calculate the remaining trigonometric functions. Use \( \sin \theta \) and \( \cos \theta \) to find \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), \( \csc \theta = \frac{1}{\sin \theta} \), and \( \cot \theta = \frac{1}{\tan \theta} \).>

Key Concepts

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The primary functions include sine, cosine, tangent, secant, cosecant, and cotangent. Understanding these functions is essential for solving problems involving angles and their relationships, particularly in right triangles.

Secant and Tangent Relationships

Secant (sec) is the reciprocal of cosine, defined as sec θ = 1/cos θ. The tangent (tan) function is the ratio of sine to cosine, tan θ = sin θ/cos θ. Given sec θ = -3, we can derive the cosine value and subsequently find the sine value, which helps in determining other trigonometric functions.

Quadrants and Signs of Trigonometric Functions

The signs of trigonometric functions depend on the quadrant in which the angle θ lies. In this case, since sec θ is negative and tan θ is positive, θ must be in the third quadrant, where sine is negative and cosine is negative, while tangent is positive. This understanding is crucial for determining the exact values of the remaining trigonometric functions.