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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 are equal because tan⁡(θ)=1\tan(\theta) = 1 means sin⁡(θ)=cos⁡(θ)\sin(\theta) = \cos(\theta).
  2. Find Angle in First Quadrant: In the first quadrant, tan⁡(θ)=1\tan(\theta) = 1 at θ=45∘\theta = 45^\circ because sin⁡(45∘)=cos⁡(45∘)=2/2\sin(45^\circ) = \cos(45^\circ) = \sqrt{2}/2.
  3. Consider First Quadrant Only: In the third quadrant, tan⁡(θ)=1\tan(\theta) = 1 would also occur, but since we're looking for −90∘<θ<90∘-90^\circ < \theta < 90^\circ, we only consider the first quadrant.
  4. Final Solution: Therefore, the only solution within the given interval is θ=45∘\theta = 45^\circ.

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