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Find all solutions with −90∘<θ<90∘-90^\circ < \theta < 90^\circ. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.\newlinetan⁡(θ)=−1\tan (\theta) = -1\newline____ ∘\_\_\_\_\,^\circ

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Full solution

Q. Find all solutions with −90∘<θ<90∘-90^\circ < \theta < 90^\circ. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.\newlinetan⁡(θ)=−1\tan (\theta) = -1\newline____ ∘\_\_\_\_\,^\circ
  1. Identify Tangent Relationship: Since tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}, we're looking for angles where the sine and cosine have the same absolute value but opposite signs.
  2. Determine Angle for Tangent=−1-1: The tangent of an angle is −1-1 when the angle is 45°45° below the x-axis in the fourth quadrant or 45°45° above the x-axis in the second quadrant.
  3. Calculate Angle in Second Quadrant: In the second quadrant, the angle is 180°−45°180° - 45°, which is 135°135°. But this is outside our range of −90°<θ<90°-90° < \theta < 90°.
  4. Calculate Angle in Fourth Quadrant: In the fourth quadrant, the angle is −45∘-45^\circ, which is within our range.
  5. Final Solution: So the only solution within the given range is θ=−45∘\theta = -45^\circ.

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