Find cos(s + t) and cos(s - t).
cos s = - 8/17 and cos t = - 3/5, s and t in quadrant III

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Identify the given information: \(\cos s = -\frac{8}{17}\) and \(\cos t = -\frac{3}{5}\), with both angles \(s\) and \(t\) in quadrant III. In quadrant III, both sine and cosine are negative.

Use the Pythagorean identity to find \(\sin s\) and \(\sin t\). Recall that \(\sin^2 \theta + \cos^2 \theta = 1\). So, calculate \(\sin s = -\sqrt{1 - \cos^2 s}\) and \(\sin t = -\sqrt{1 - \cos^2 t}\) because sine is negative in quadrant III.

Write down the cosine addition and subtraction formulas: \(\cos(s + t) = \cos s \cos t - \sin s \sin t\) \(\cos(s - t) = \cos s \cos t + \sin s \sin t\)

Substitute the known values of \(\cos s\), \(\cos t\), \(\sin s\), and \(\sin t\) into the formulas for \(\cos(s + t)\) and \(\cos(s - t)\).

Simplify the expressions by performing the multiplications and combining like terms to express \(\cos(s + t)\) and \(\cos(s - t)\) in terms of fractions.

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Key Concepts

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

Cosine Addition and Subtraction Formulas

These formulas express the cosine of a sum or difference of two angles in terms of the cosines and sines of the individual angles: cos(s + t) = cos s cos t - sin s sin t and cos(s - t) = cos s cos t + sin s sin t. They are essential for breaking down complex angle expressions into known values.

Sign and Value of Trigonometric Functions in Quadrant III

In the third quadrant, both sine and cosine values are negative. Knowing the quadrant helps determine the correct signs of sine and cosine when calculating unknown values, such as sin s and sin t, from given cosine values using the Pythagorean identity.

Pythagorean Identity

The identity sin²θ + cos²θ = 1 allows calculation of sine values when cosine values are known. Given cos s and cos t, sin s and sin t can be found by rearranging the identity, considering the sign based on the quadrant, which is crucial for applying the addition and subtraction formulas.

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