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If sin⁡(t)=78\sin(t) = \frac{7}{8} and tt is in the 22nd quadrant, find cos⁡(t)\cos(t).

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Full solution

Q. If sin⁡(t)=78\sin(t) = \frac{7}{8} and tt is in the 22nd quadrant, find cos⁡(t)\cos(t).
  1. Apply Pythagorean Identity: Use the Pythagorean identity for sine and cosine.\newlineThe Pythagorean identity states that sin⁡2(t)+cos⁡2(t)=1\sin^2(t) + \cos^2(t) = 1.
  2. Substitute given value: Substitute the given value of sin⁡(t)\sin(t) into the identity.sin⁡(t)=78\sin(t) = \frac{7}{8}, so sin⁡2(t)=(78)2=4964\sin^2(t) = \left(\frac{7}{8}\right)^2 = \frac{49}{64}.
  3. Solve for cos⁡2(t)\cos^2(t): Solve for cos⁡2(t)\cos^2(t).cos⁡2(t)=1−sin⁡2(t)=1−4964=6464−4964=1564\cos^2(t) = 1 - \sin^2(t) = 1 - \frac{49}{64} = \frac{64}{64} - \frac{49}{64} = \frac{15}{64}.
  4. Determine cosine sign: Determine the sign of cos⁡(t)\cos(t) in the 22nd quadrant.\newlineIn the 22nd quadrant, cosine is negative. Therefore, cos⁡(t)\cos(t) is negative.
  5. Find cos⁡(t)\cos(t): Take the square root of cos⁡2(t)\cos^2(t) to find cos⁡(t)\cos(t).cos⁡(t)=−1564=−158\cos(t) = -\sqrt{\frac{15}{64}} = -\frac{\sqrt{15}}{8}.

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