Problem 53 - Precalculus

Question

For the following exercises, rewrite in terms of \(\sin x\) and \(\cos x .\) $$ \sin \left(x-\frac{3 \pi}{4}\right) $$

Step-by-Step Solution

Answer

\( \sin\left(x-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \sin x - \frac{\sqrt{2}}{2} \cos x \)

Step 1: Identify the Trigonometric Identity

We will use the angle subtraction identity for sine. The formula is: \[ \sin(a - b) = \sin a \cos b - \cos a \sin b\] In this case, we'll consider \(a = x\) and \(b = \frac{3\pi}{4}\).

Step 2: Apply the Trigonometric Identity

Using the identity from Step 1, replace \(a\) and \(b\) with the given values. This gives us: \[ \sin \left(x-\frac{3 \pi}{4}\right) = \sin x \cdot \cos \left(\frac{3\pi}{4}\right) - \cos x \cdot \sin \left(\frac{3\pi}{4}\right) \]

Step 3: Calculate Known Trig Values

Now we need to calculate \(\cos\left(\frac{3\pi}{4}\right)\) and \(\sin\left(\frac{3\pi}{4}\right)\). - Since \(\frac{3\pi}{4}\) is in the second quadrant where cosine is negative and sine is positive: \[ \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}, \quad \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \]

Step 4: Substitute the Known Values

Substitute the values calculated in Step 3 into the expression derived in Step 2: \[ \sin\left(x-\frac{3\pi}{4}\right) = \sin x \cdot \left(-\frac{\sqrt{2}}{2}\right) - \cos x \cdot \frac{\sqrt{2}}{2} \] Simplifying this gives: \[ \sin\left(x-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \sin x - \frac{\sqrt{2}}{2} \cos x \]

Key Concepts

Angle Subtraction Identity

The angle subtraction identity is a useful tool when working with trigonometric functions like sine and cosine. This identity allows you to express the sine and cosine of a difference between two angles in terms of the sines and cosines of the individual angles. For sine, the angle subtraction identity is formulated as:

• \( \sin(a - b) = \sin a \cos b - \cos a \sin b \)

Here, \( a \) and \( b \) are the angles involved. Understanding this identity is crucial because it helps to simplify complex trigonometric expressions and solve related equations.
For example, in the exercise provided, \( a \) is set as \( x \) and \( b \) as \( \frac{3\pi}{4} \). By applying the identity, you're able to break down the expression \( \sin \left(x-\frac{3 \pi}{4}\right) \) into components of \( \sin x \) and \( \cos x \), which are easier to work with.

Sine Function

The sine function is a fundamental part of trigonometry, revolving around the properties of angles in triangles. It is denoted by \( \sin \) and is one of the basic trigonometric functions, the others being cosine and tangent. The sine of an angle \( \theta \) in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse.

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)

The sine function is also significant in the unit circle, where any angle corresponds to a point on the circle, and the y-coordinate of that point is the sine of the angle.
Beyond right triangles, the sine function has applications in wave motion, harmonic oscillations, and other periodic phenomena. In specific situations, like in the original exercise, it is important to apply identities like the angle subtraction identity to work with sine in a more manageable form.

Cosine Function

The cosine function, denoted as \( \cos \), is closely related to the sine function. In a right triangle, the cosine of an angle \( \theta \) is the ratio of the length of the adjacent side to the hypotenuse.

• \( \cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} \)

On the unit circle, the cosine of an angle corresponds to the x-coordinate of the point representing that angle.
Cosine is especially useful when combined with sine through identities like the angle subtraction identity. It is crucial in determining the phase, amplitude, and structure of trigonometric expressions. In the exercise, \( \frac{3\pi}{4} \) places the angle in the second quadrant, where cosine values are negative. This affects the calculation and final simplification when applying the angle subtraction identity.