Use the given information to find tan(s + t). See Example 3.
cos s = -15/17 and sin t = 4/5, s in quadrant II and t in quadrant I

Verified step by step guidance

Identify the given information: \( \cos s = -\frac{15}{17} \) with \( s \) in quadrant II, and \( \sin t = \frac{4}{5} \) with \( t \) in quadrant I.

Find \( \sin s \) using the Pythagorean identity \( \sin^2 s + \cos^2 s = 1 \). Since \( s \) is in quadrant II, \( \sin s \) is positive. Calculate \( \sin s = \sqrt{1 - \cos^2 s} = \sqrt{1 - \left(-\frac{15}{17}\right)^2} \).

Find \( \cos t \) using the Pythagorean identity \( \sin^2 t + \cos^2 t = 1 \). Since \( t \) is in quadrant I, \( \cos t \) is positive. Calculate \( \cos t = \sqrt{1 - \sin^2 t} = \sqrt{1 - \left(\frac{4}{5}\right)^2} \).

Use the angle addition formula for tangent: \[ \tan(s + t) = \frac{\tan s + \tan t}{1 - \tan s \tan t} \]. To apply this, find \( \tan s = \frac{\sin s}{\cos s} \) and \( \tan t = \frac{\sin t}{\cos t} \).

Substitute the values of \( \tan s \) and \( \tan t \) into the formula and simplify the expression to find \( \tan(s + t) \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Identities for Sum of Angles

The tangent of the sum of two angles, tan(s + t), can be found using the identity tan(s + t) = (tan s + tan t) / (1 - tan s * tan t). This formula allows combining individual tangent values to find the tangent of their sum.

Determining Sine, Cosine, and Tangent from Given Information

Given cos s and sin t along with their quadrants, you can find the missing sine or cosine values using the Pythagorean identity sin²θ + cos²θ = 1. The quadrant information helps determine the sign of these values, which is crucial for accurate tangent calculation.

Sign of Trigonometric Functions in Different Quadrants

The signs of sine, cosine, and tangent depend on the quadrant of the angle. In quadrant II, sine is positive and cosine is negative; in quadrant I, both sine and cosine are positive. Correctly assigning signs ensures the correct evaluation of tan(s + t).

Recommended video: