Problem 64 - Precalculus with Limits

Question

In Exercises 61 - 70, prove the identity. \( \cos\left(\dfrac{5\pi}{4} - x\right) = - \dfrac{\sqrt{2}}{2}\left(\cos x + \sin x\right) \)

Step-by-Step Solution

Answer

The given identity \(\cos\left(\dfrac{5\pi}{4} - x\right) = - \dfrac{\sqrt{2}}{2}\left(\cos x + \sin x\right)\) is proven with using the properties and values of cosine and sine.

Step 1: Use the formula for cos(A - B)

Apply the formula for \(\cos (A - B) = \cos A \cos B + \sin A \sin B\) to given identity \(\cos\left(\dfrac{5\pi}{4} - x\right)\). Therefore, this becomes \(\cos(\frac{5 \pi}{4})\cos(x) + \sin(\frac{5 \pi}{4})\sin(x)\).

Step 2: Substitute the values of cos and sin

Substitute \(\cos(\frac{5 \pi}{4})\) and \(\sin(\frac{5 \pi}{4})\) with their values which are \(-\frac{\sqrt{2}}{2}\). So, it becomes \(-\frac{\sqrt{2}}{2}\cos(x) - \frac{\sqrt{2}}{2}\sin(x)\).

Step 3: Factor out \(-\frac{\sqrt{2}}{2}\)

Bring \(-\frac{\sqrt{2}}{2}\) out of the brackets. This gives \(-\frac{\sqrt{2}}{2}(\cos(x) + \sin(x))\).

Key Concepts

cosine difference formula

To understand the problem at hand, it is essential to grasp the **Cosine Difference Formula**. This formula expresses the cosine of the difference between two angles, A and B. It is written as \( \cos(A - B) = \cos A \cos B + \sin A \sin B \). This trigonometric identity is invaluable for simplifying expressions and proving equations where angles are subtracted.

The Cosine Difference Formula works because, in trigonometry, the functions \( \cos \) and \( \sin \) relate to the X and Y coordinates of a point on a unit circle. By understanding this relationship, mathematicians have derived this formula to simplify complex trigonometric expressions. In our exercise, we directly apply this formula to \( \cos(\frac{5\pi}{4} - x) \) to start breaking down the identity. This step is pivotal as it allows us to delve further into the problem by replacing the angle subtraction with known trigonometric values.

Remembering and utilizing the Cosine Difference Formula can greatly simplify calculations and is a fundamental tool in trigonometry.

trigonometric functions

Trigonometric functions are core components in trigonometry, involving the functions \( \cos \), \( \sin \), and \( \tan \), among others. These functions relate the angles of triangles to the lengths of their sides, especially in right angles. They're not only critical in theoretical mathematics but also practical applications such as physics, engineering, and even computer graphics.

In our exercise, we focus on the cosine and sine trigonometric functions for the angle \( \frac{5\pi}{4} \). This specific angle is located in the third quadrant of the unit circle where both sine and cosine values are negative. For \( \frac{5\pi}{4} \), both \( \cos \) and \( \sin \) are \( -\frac{\sqrt{2}}{2} \).

• The cosine function deals with the adjacent side and hypotenuse in a right triangle.
• The sine function relates to the opposite side and hypotenuse.
• Together, they help in understanding the behavior of angles and their trigonometric identities.

By substituting these known trigonometric values, you can begin simplifying or resolving trigonometric expressions, as demonstrated in the original exercise.

angle subtraction identities

Angle subtraction identities are crucial tools in trigonometry, allowing mathematicians to evaluate trigonometric functions of angles that don't fit nicely into simple, measurable parts of the circle like \( \pi/4 \) or \( \pi/2 \). Instead, these identities help break down more complex angles into components that can be simplified using known trigonometric functions.

In our problem, the angle subtraction identity at play is used to split \( \cos(\frac{5 \pi}{4} - x) \) into multiple parts involving \( \cos \) and \( \sin \) functions that correlate with distinct, recognizable angles. This identity allows us to take the unfamiliar and transform it into a recognizably simple trigonometric equation.

By applying the angle subtraction identity, one can:

• Use known values of standard trigonometric functions to simplify expressions.
• Break down complex angles into the sum or difference of simpler angles.
• Model real-world phenomena involving periodic or wave patterns, thanks to their modular nature.

Angle subtraction identities are, therefore, a cornerstone in not only proving identities as shown in the solution but also in various complex mathematical and applied science problems.