Question

Write each expression as a single trigonometric function. a) \(\cos 43^{\circ} \cos 27^{\circ}-\sin 43^{\circ} \sin 27^{\circ}\) b) \(\sin 15^{\circ} \cos 20^{\circ}+\cos 15^{\circ} \sin 20^{\circ}\) c) \(\cos ^{2} 19^{\circ}-\sin ^{2} 19^{\circ}\) d) \(\sin \frac{3 \pi}{2} \cos \frac{5 \pi}{4}-\cos \frac{3 \pi}{2} \sin \frac{5 \pi}{4}\) e) \(8 \sin \frac{\pi}{3} \cos \frac{\pi}{3}\)

Short answer

a) \(\cos 16^{\circ}\) b) \(\sin 35^{\circ}\) c) \(\cos 38^{\circ}\) d) \(\frac{\sqrt{2}}{2}\) e) \(4 \sin 120^{\circ}\)

Step-by-step solution

Identify Trigonometric Identities

Use known trigonometric identities to rewrite each expression as a single trigonometric function. The main identities used here are the angle addition and subtraction formulas.

Simplify Part (a)

Given: \[\cos 43^{\circ} \cos 27^{\circ}-\sin 43^{\circ} \sin 27^{\circ}\]This can be simplified using the cosine of the sum formula: \[\cos(A - B) = \cos A \cos B - \sin A \sin B\]Here, \(A = 43^{\circ}\) and \(B = 27^{\circ}\). Thus, \[\cos(43^{\circ} - 27^{\circ}) = \cos 16^{\circ}\]

Simplify Part (b)

Given: \[\sin 15^{\circ} \cos 20^{\circ} + \cos 15^{\circ} \sin 20^{\circ}\]This can be simplified using the sine of the sum formula: \[\sin(A + B) = \sin A \cos B + \cos A \sin B\]Here, \(A = 15^{\circ}\) and \(B = 20^{\circ}\). Thus, \[\sin (15^{\circ} + 20^{\circ}) = \sin 35^{\circ}\]

Simplify Part (c)

Given: \[\cos^{2} 19^{\circ} - \sin^{2} 19^{\circ}\]This can be simplified using the cosine double angle formula: \[\cos (2A) = \cos^{2} A - \sin^{2} A\]Here, \(A = 19^{\circ}\). Thus, \[\cos (2 \times 19^{\circ}) = \cos 38^{\circ}\]

Simplify Part (d)

Given: \[\sin \frac{3 \pi}{2} \cos \frac{5 \pi}{4} - \cos \frac{3 \pi}{2} \sin \frac{5 \pi}{4}\]This can be simplified using the sine of the difference formula: \[\sin(A - B) = \sin A \cos B - \cos A \sin B\]Here, \(A = \frac{3 \pi}{2}\) and \(B = \frac{5 \pi}{4}\). Thus, \[\sin \left( \frac{3 \pi}{2} - \frac{5 \pi}{4} \right) = \sin \left( \frac{6 \pi - 5 \pi}{4} \right) = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\]

Simplify Part (e)

Given: \[8 \sin \frac{\pi}{3} \cos \frac{\pi}{3}\]This can be simplified using the sine double angle formula: \[\sin(2A) = 2 \sin A \cos A\]Here, \(A = \frac{\pi}{3}\). Thus, \[8 \sin \frac{\pi}{3} \cos \frac{\pi}{3} = 8 \times \frac{1}{2} \sin(2 \times \frac{\pi}{3}) = 4 \sin \frac{2\pi}{3} = 4 \sin 120^{\circ}\]

Key concepts

Angle Addition Formula

The angle addition formula is crucial in simplifying trigonometric expressions. It states that for any angles A and B, the sine and cosine of the sum of the two angles can be expressed as:
\(\begin{align*}\sin(A + B) &= \sin A \cos B + \cos A \sin B \cos(A + B) &= \cos A \cos B - \sin A \sin Bd\end{align*}\)
The formula transforms the sum of trigonometric functions into the product of sines and cosines. This helps in handling and resolving trigonometric expressions efficiently. For example, in Step 3 of the solution, we used \(\sin(15^{\circ} + 20^{\circ}) = \sin 35^{\circ}\) by applying the angle addition formula.

Angle Subtraction Formula

Similar to the angle addition formula, the angle subtraction formula is important for manipulating trigonometric expressions. It states that for any angles A and B:
\(\begin{align*}\sin(A - B) &= \sin A \cos B - \cos A \sin B \cos(A - B) &= \cos A \cos B + \sin A \sin B\end{align*}\)
This formula allows us to express the subtraction of trigonometric functions in terms of multiplication. For instance, in Step 2, we simplified \(\cos 43^{\circ} \cos 27^{\circ} - \sin 43^{\circ} \sin 27^{\circ}\) to \(\cos(43^{\circ} - 27^{\circ}) = \cos 16^{\circ}\), by using the cosine angle subtraction formula.

Cosine Double Angle Formula

The cosine double angle formula is especially useful when dealing with expressions involving squared sine and cosine terms. It states:
\(\begin{align*}\cos(2A) &= \cos^2 A - \sin^2 A \cos(2A) &= 2 \cos^2 A - 1 \cos(2A) &= 1 - 2 \sin^2 A\end{align*}\)
Each of these forms can be used depending on the given problem to simplify expressions effectively. In Step 4, we used \(\cos 2A = \cos^2 A - \sin^2 A\) to rewrite \(\cos^2 19^{\circ} - \sin^2 19^{\circ}\) as \(\cos(2 \times 19^{\circ}) = \cos 38^{\circ}\).

Sine Double Angle Formula

The sine double angle formula is an identity that makes it easier to handle products of sine and cosine terms. It states:
\(\sin(2A) = 2 \sin A \cos A\)
This formula is helpful for simplifying the product of a sine and cosine term into a single sine term of double the angle. In Step 6, we transformed \(8 \sin \frac{\pi}{3} \cos \frac{\pi}{3}\) into \(4 \sin 120^{\circ}\) using this formula, since \(8 \sin \frac{\pi}{3} \cos \frac{\pi}{3} = 4 \sin 120^{\circ}\).