Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Use identities to write each expression as a function with \(x\) as the only argument. $$\cos \left(135^{\circ}-x\right)$$

Step-by-Step Solution

Answer

\( \cos(135^{\circ}-x) = -\frac{\sqrt{2}}{2} \cos x + \frac{\sqrt{2}}{2} \sin x \).

Step 1: Use the Cosine Subtraction Identity

The cosine of a difference can be rewritten using the identity: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \). For this problem, let \( a = 135^{\circ} \) and \( b = x \).

Step 2: Compute Known Trigonometric Values

We need to find \( \cos 135^{\circ} \) and \( \sin 135^{\circ} \). - Since \( 135^{\circ} = 180^{\circ} - 45^{\circ} \), and it is in the second quadrant, where cosine is negative and sine is positive, we have: - \( \cos 135^{\circ} = -\frac{\sqrt{2}}{2} \) - \( \sin 135^{\circ} = \frac{\sqrt{2}}{2} \)

Step 3: Substitute Values Back into the Identity

Substitute \( \cos 135^{\circ} = -\frac{\sqrt{2}}{2} \) and \( \sin 135^{\circ} = \frac{\sqrt{2}}{2} \) into the identity from Step 1:\[ \cos(135^{\circ} - x) = \left(-\frac{\sqrt{2}}{2}\right) \cos x + \left(\frac{\sqrt{2}}{2}\right) \sin x \].

Step 4: Simplify the Expression

Now, the expression \( \cos(135^{\circ} - x) \) simplifies directly to:\[ \cos(135^{\circ} - x) = -\frac{\sqrt{2}}{2} \cos x + \frac{\sqrt{2}}{2} \sin x \].

Key Concepts

Cosine Subtraction Identity

When working with trigonometric expressions, identities can streamline calculations by transforming one form into another. The cosine subtraction identity is useful for expressing the cosine of a difference between two angles. It is presented as:

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

This identity tells us how to break down a complex angle into simpler parts. In the given exercise, the problem involves calculating \( \cos(135^\circ - x) \).
By recognizing \( a = 135^\circ \) and \( b = x \), we can apply the identity to rewrite the cosine expression. This not only simplifies the evaluation but also transforms the expression into a form that is easier to manage and solve.
Understanding how to use the cosine subtraction identity is fundamental when manipulating trigonometric equations. It can convert more complex expressions into basic trigonometric form, aiding both mathematical comprehension and problem-solving.

Second Quadrant Trigonometric Values

When evaluating trigonometric functions, it is important to consider the quadrant in which the angle lies. The unit circle helps us determine the sign of trigonometric values. For example, when an angle like \( 135^\circ \) is analyzed, it falls within the second quadrant.
In the second quadrant:

• Cosine values are negative.
• Sine values are positive.

To find \( \cos 135^\circ \) and \( \sin 135^\circ \), recognize that \( 135^\circ \) can be expressed as \( 180^\circ - 45^\circ \). This approach simplifies the calculation:

• \( \cos 135^\circ = -\frac{\sqrt{2}}{2} \)
• \( \sin 135^\circ = \frac{\sqrt{2}}{2} \)

Knowing these second quadrant values helps us correctly apply and compute expressions involving these angles. Familiarity with the characteristics of angles in each quadrant enables efficient problem-solving.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves substituting known values and applying identities to decrease complexity. After determining the cosine and sine of \( 135^\circ \), substitute these into the earlier cosine subtraction identity.
The expression \( \cos(135^\circ - x)\) becomes:

• \(-\frac{\sqrt{2}}{2} \cos x + \frac{\sqrt{2}}{2} \sin x \)

This illustrates a simplified form, which is easier to interpret and solve. Simplifying expressions using identities makes it more straightforward to understand and solve equations, highlighting relationships within trigonometric functions.
It's helpful to memorize certain trigonometric identities and values from the unit circle. This knowledge not only enables the simplification of expressions but also assists in more complex trigonometric problem-solving scenarios. Simplification is a key skill in mathematics, reducing errors and enhancing procedural clarity.