Find cos(s + t) and cos(s - t).
cos s = √2/4 and sin t = - √5/6, s and t in quadrant IV

검증된 단계별 안내

Identify the given information: \(\cos s = \frac{\sqrt{2}}{4}\) and \(\sin t = -\frac{\sqrt{5}}{6}\), with both angles \(s\) and \(t\) in quadrant IV.

Since \(s\) and \(t\) are in quadrant IV, recall that in quadrant IV, cosine is positive and sine is negative. Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\sin s\) and \(\cos t\).

Calculate \(\sin s\) using \(\sin s = -\sqrt{1 - \cos^2 s}\) because sine is negative in quadrant IV. Similarly, calculate \(\cos t = \sqrt{1 - \sin^2 t}\) because cosine is positive in quadrant IV.

Use 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 \(\cos s\), \(\sin s\), \(\cos t\), and \(\sin t\) into the formulas to express \(\cos(s + t)\) and \(\cos(s - t)\) in terms of known quantities.

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주요 개념

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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 Sine and Cosine Values in Specific Quadrants

Knowing the quadrant of an angle helps determine the sign of its sine and cosine values. In quadrant IV, cosine is positive and sine is negative. This information is crucial for correctly assigning signs to trigonometric values when calculating unknown functions.

Using Pythagorean Identity to Find Missing Trigonometric Values

The Pythagorean identity, sin²θ + cos²θ = 1, allows calculation of an unknown sine or cosine value when the other is known. Applying this identity with the correct sign based on the quadrant helps find missing values needed to evaluate expressions like cos(s + t) and cos(s - t).

Find cos(s + t) and cos(s - t).cos s = √2/4 and sin t = - √5/6, s and t in quadrant IV