Triple-Angle Identities
By substituting \(b = 2a\) into the sum identity \(\sin (a + b) = \sin a \cos b + \sin b \cos a\), we derive:
\[ 1) \ \sin 3a = 3 \sin a – 4 \sin^3 a \]
By substituting \(b = 2a\) into the sum identity \(\cos (a + b) = \cos a \cos b – \sin a \sin b\), we derive:
\[ 2) \ \cos 3a = 4 \cos^3 a – 3 \cos a \]
By substituting \(b = 2a\) into the sum identity:
\[ \tan (a + b) = \frac{\tan a + \tan b}{1 – \tan a \tan b} \]
we derive:
\[ 3) \ \tan 3a = \frac{3 \tan a – \tan^3 a}{1 – 3 \tan^2 a} \]
Example:
Given \(\tan a = \frac{4}{3}\), find the value of \(\cos 3a\).
If \(\tan a = \frac{4}{3}\), then by constructing a right triangle, we find \(\cos a = \frac{3}{5}\). Using the triple-angle identity:
\[ \cos 3a = 4 \cos^3 a – 3 \cos a \]
\[ = 4 \cdot \left( \frac{3}{5} \right)^3 – 3 \cdot \frac{3}{5} = -\frac{117}{125} \]
Question 34
If \(\cos 7^\circ = a\), express the statement \(\sin 69^\circ + 3 \sin 83^\circ\) in terms of \(a\).
\[ A) -3a \quad B) -4a^3 \quad C) 4a^3 \quad D) 3a \quad E) a^3 \]
Solution:
Using the cofunction identities \(\sin \theta = \cos(90^\circ – \theta)\):
\[ \sin 69^\circ + 3 \sin 83^\circ = \cos 21^\circ + 3 \cos 7^\circ \]
Since \(21^\circ = 3(7^\circ)\), we substitute the triple-angle identity for \(\cos 21^\circ\):
\[ = (4 \cos^3 7^\circ – 3 \cos 7^\circ) + 3 \cos 7^\circ \]
\[ = 4 \cos^3 7^\circ = 4a^3 \]
\(\text{Correct Answer: C} \)