Find cos(s + t) and cos(s - t).
sin s = 2/3 and sin t = -1/3, s in quadrant II and t in quadrant IV

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Identify the given information: \(\sin s = \frac{2}{3}\) with \(s\) in quadrant II, and \(\sin t = -\frac{1}{3}\) with \(t\) in quadrant IV. Recall that in quadrant II, cosine is negative, and in quadrant IV, cosine is positive.

Use the Pythagorean identity to find \(\cos s\) and \(\cos t\). Since \(\sin^2 \theta + \cos^2 \theta = 1\), calculate \(\cos s = -\sqrt{1 - \sin^2 s}\) (negative because \(s\) is in quadrant II) and \(\cos t = +\sqrt{1 - \sin^2 t}\) (positive because \(t\) is in quadrant IV).

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

Substitute the values of \(\sin s\), \(\sin t\), \(\cos s\), and \(\cos t\) 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 known values.

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Key Concepts

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

Trigonometric Angle Sum and Difference Formulas

These formulas express the cosine of the sum or difference of two angles in terms of the sines and cosines 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.

Determining Cosine from Sine Using Quadrants

Given sin s and sin t along with their quadrants, we use the Pythagorean identity cos²θ = 1 - sin²θ to find cos s and cos t. The sign of cosine depends on the quadrant: in quadrant II, cosine is negative; in quadrant IV, cosine is positive.

Sign Conventions in Different Quadrants

The signs of sine and cosine vary by quadrant: in quadrant II, sine is positive and cosine is negative; in quadrant IV, sine is negative and cosine is positive. Correctly applying these sign rules ensures accurate calculation of trigonometric values.

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