Problem 38 - Precalculus: Functions and Graphs

Question

Exer. \(37-46:\) Verify the identity. $$\cos \left(\theta+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\cos \theta-\sin \theta)$$

Step-by-Step Solution

Answer

The identity is verified as true by using the cosine angle addition formula.

Step 1: Recall the Angle Addition Formula for Cosine

The angle addition formula for cosine is given by: \(\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b)\). For this problem, we have \(a = \theta\) and \(b = \frac{\pi}{4}\).

Step 2: Calculate \(\cos\left(\frac{\pi}{4}\right)\) and \(\sin\left(\frac{\pi}{4}\right)\)

The values for the trigonometric functions at \(\frac{\pi}{4}\) are: \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\) and \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).

Step 3: Substitute Values into the Formula

Substitute the values from Step 2 into the angle addition formula: \[\cos\left(\theta+\frac{\pi}{4}\right) = \cos\theta \cdot \frac{\sqrt{2}}{2} - \sin\theta \cdot \frac{\sqrt{2}}{2}\]This simplifies to: \[\frac{\sqrt{2}}{2}(\cos\theta - \sin\theta)\]

Step 4: Verify the Identity

Notice that the expression obtained from the angle addition formula \(\frac{\sqrt{2}}{2}(\cos\theta - \sin\theta)\) matches the right side of the given equation \(\cos(\theta+\frac{\pi}{4}) = \frac{\sqrt{2}}{2}(\cos\theta - \sin\theta)\). Therefore, the identity is verified.

Key Concepts

Angle Addition Formula

The angle addition formula is a powerful tool in trigonometry that simplifies the calculation of trigonometric functions involving sums of angles. For the cosine function, the formula is given by \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \). This formula allows us to find the cosine of the sum of two angles by using the trigonometric values of each component angle separately. In our exercise, we use this formula with \( a = \theta \) and \( b = \frac{\pi}{4} \). This step simplifies the calculation by breaking down the problem into manageable parts. Using known values for \( \cos \) and \( \sin \) at \( \frac{\pi}{4} \), we can easily substitute and check the given trigonometric identity.

Cosine Function

The cosine function is one of the primary trigonometric functions, measuring the horizontal component of a point on the unit circle corresponding to a given angle. It plays a crucial role in many areas of mathematics, especially in verifying identities like our current exercise. In the angle addition formula \( \cos(a + b) \), the cosine function's properties allow us to express the cosine of an angle sum in terms of the cosine and sine of the individual angles. For \( \frac{\pi}{4} \) radians, the cosine value is \( \frac{\sqrt{2}}{2} \). This symmetric trigonometric ratio is fundamental in simplifying complex expressions and often appears in problems involving 45-degree angles or equivalent radians.

Trigonometric Functions

Trigonometric functions encompass the relationships between the angles and sides of triangles. The primary trigonometric functions — sine, cosine, and tangent — have specific roles in solving various geometry and physics problems. They are identified by their unique cyclical patterns and ratios. The sine and cosine functions are particularly important for their roles in circular motion and wave patterns. For example, in our exercise, the \( \sin \) and \( \cos \) values for \( \frac{\pi}{4} \) both equal \( \frac{\sqrt{2}}{2} \). This symmetric property underpins the beauty and elegance often found in trigonometric identities, where angles relate in simple and predictable patterns.