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4.1 — Radian Measure
Question 1
60° converted to radians is:
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Solution:
\( 60° \times \frac{\pi}{180°} = \frac{60\pi}{180} = \frac{\pi}{3} \)
Question 2
\( \frac{5\pi}{4} \) radians converted to degrees is:
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Solution:
\( \frac{5\pi}{4} \times \frac{180°}{\pi} = \frac{5 \times 180}{4} = 225° \)
4.2 — Special Angles
Question 3
\( \sin\left(\frac{\pi}{3}\right) = \)
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Solution:
\( \sin(60°) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \) from the 30-60-90 special triangle.
Question 4
\( \cos\left(\frac{5\pi}{4}\right) = \)
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Solution:
\( \frac{5\pi}{4} \) is in Q3 (225°). Reference angle = \( \frac{\pi}{4} \). Cosine is negative in Q3.
\( \cos\left(\frac{5\pi}{4}\right) = -\cos\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)
4.4 — Compound Angles
Question 5
The exact value of \( \sin(75°) \) using \( \sin(45°+30°) \) is:
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Solution:
\( \sin(45°+30°) = \sin 45°\cos 30° + \cos 45°\sin 30° \)
\( = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6}+\sqrt{2}}{4} \)
MHF4U — Advanced Functions, Grade 12 | Ontario Curriculum 2007 (Revised)