Question
S use sin Rule and cosine Rule, Identitiogs following without 9 to evaluate the for calculator D \( \sin 15^{\prime} \cos 15^{\circ} \)
Ask by Washington Ayala. in South Africa
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\[
\sin 15^\circ \cos 15^\circ = \frac{1}{4}
\]
Solution
We start with the product
\[
\sin 15^\circ \cos 15^\circ.
\]
A useful identity for the product of \(\sin\) and \(\cos\) of the same angle is
\[
\sin \theta \cos \theta = \frac{1}{2} \sin (2\theta).
\]
Taking \(\theta = 15^\circ\), we get
\[
\sin 15^\circ \cos 15^\circ = \frac{1}{2} \sin (2 \times 15^\circ) = \frac{1}{2} \sin 30^\circ.
\]
We know that
\[
\sin 30^\circ = \frac{1}{2}.
\]
Thus,
\[
\frac{1}{2} \sin 30^\circ = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}.
\]
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Simplify this solution Extra Insights
To evaluate \(\sin 15^{\circ} \cos 15^{\circ}\) without a calculator, you can use the double angle identity for sine. This identity states that \(\sin(2\theta) = 2 \sin(\theta) \cos(\theta)\). Therefore, \(\sin 15^{\circ} \cos 15^{\circ} = \frac{1}{2} \sin(30^{\circ}) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}\). Another way to verify this is by using the angle subtraction identity. You know that \( \sin(30^{\circ}) = \frac{1}{2} \) and you can express \( \sin(15^{\circ}) \) and \( \cos(15^{\circ}) \) in terms of \( \sin(30^{\circ}) \) from the right triangle ratios and the unit circle. Connect it all together and you’ll enjoy the magic of trigonometry!
