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If tan theta = sqrt(3), find (i) the size of theta (ii) sin theta

Mathematics

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Answers

2 parts
i

60∘\text{60}^\circ

ii

32\frac{\sqrt{3}}{2}

Step-by-step solution

4 steps

Here's how to solve the problem:

Part i· 2 steps

The size of angle θ\theta

  1. Use the given tangent value to find θ\theta.

    We are given tan⁡θ=3\tan \theta = \sqrt{3}. To find θ\theta, we take the inverse tangent of both sides.

    θ=arctan⁡(3)\theta = \arctan(\sqrt{3})
  2. Calculate the value of θ\theta.

    We know that tan⁡(60∘)=3\tan(60^\circ) = \sqrt{3}.

    θ=60∘\theta = 60^\circ

    The size of angle θ\theta is 60∘\boxed{\text{60}^\circ}.

Part ii

32\frac{\sqrt{3}}{2}

sin⁡θ\sin \theta

  1. Substitute the value of θ\theta into the expression.

    Now that we know θ=60∘\theta = 60^\circ, we need to find sin⁡(60∘)\sin(60^\circ).

    sin⁡θ=sin⁡(60∘)\sin \theta = \sin(60^\circ)
  2. Calculate the sine value.

    The value of sin⁡(60∘)\sin(60^\circ) is a standard trigonometric value.

    sin⁡(60∘)=32\sin(60^\circ) = \frac{\sqrt{3}}{2}

    The value of sin⁡θ\sin \theta is 32\boxed{\frac{\sqrt{3}}{2}}.

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