tan (θ + φ); cos(θ) = −1/3, θ in Quadrant III, sin(φ) = 1/4, φ in Quadrant II. Evaluate the expression.

Solution:

Given:

cos(θ) = −1/3, θ in Quadrant III

sin(φ) = 1/4, φ in Quadrant II

To find: tan (θ + φ)

To evaluate the trigonometric expression, let us determine θ and Φ using inverse trigonometric functions.

Θ = 180° + cos⁻¹(1/3)

= 180° + 70.5°

= 250.5°

Φ = 180° - sin⁻¹(1/4)

= 180°- 14.4°

= 165.5°

tan (θ + φ) = tan( 250.5° + 165.5°)

= tan(416°)

= tan(360° +56°)

= tan(56°)

= 1.48

---

tan (θ + φ); cos(θ) = −1/3, θ in Quadrant III, sin(φ) = 1/4, φ in Quadrant II evaluate the expression.

Summary:

If cos(θ) = −1/3, θ in Quadrant III, sin(φ) = 1/4, φ in Quadrant II, then the value of tan (θ + φ) is 1.48.