Problem 12 - Algebra and Trigonometry

Question

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) Verify each identity. $$ \cos \left(x-\frac{5 \pi}{4}\right)=-\frac{\sqrt{2}}{2}(\cos x+\sin x) $$

Step-by-Step Solution

Answer

The given identity is not verified.

Step 1: Apply the formula of cosine of difference of two angles

Using the formula for the cosine of the difference of two angles \( \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \), where \( a = x \) and \( b = \frac{5 \pi}{4} \), so we can substitute these values into the formula: \( \cos \left(x-\frac{5 \pi}{4}\right) = \cos(x)\cos \left(\frac{5 \pi}{4}\right) + \sin(x)\sin\left(\frac{5 \pi}{4}\right) \)

Step 2: Calculate cosine and sine values

The values of \( \cos \left(\frac{5 \pi}{4}\right) \) and \( \sin \left(\frac{5 \pi}{4}\right) \) are \( -\frac{\sqrt{2}}{2} \) and \( \frac{\sqrt{2}}{2} \), respectively. Substitute these values into the previous step:\( \cos \left(x-\frac{5 \pi}{4}\right) = \cos(x)(-\frac{\sqrt{2}}{2}) + \sin(x)(\frac{\sqrt{2}}{2}) \)

Step 3: Simplify

Simplify the equation to see if it matches \( -\frac{\sqrt{2}}{2}(\cos x+\sin x) \):\( \cos \left(x-\frac{5 \pi}{4}\right) =-\frac{\sqrt{2}}{2}\cos x +\frac{\sqrt{2}}{2}\sin x =-\frac{\sqrt{2}}{2}(\cos x-\sin x) \)As the result doesn't match \( -\frac{\sqrt{2}}{2}(\cos x+\sin x) \), the given identity is not verified.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools used in mathematics to transform and simplify trigonometric expressions. These identities are equations involving trigonometric functions that are true for every value of the variables involved. They help in solving equations, verifying expressions, and simplifying complex mathematical problems.

Some common trigonometric identities include:

• Pythagorean Identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Angle Sum and Difference Identity: \( \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) \)
• Double Angle Identity: \( \sin(2x) = 2\sin(x)\cos(x) \)
• Reciprocal Identity: \( \csc(x) = \frac{1}{\sin(x)} \)

Understanding these identities is crucial when tackling exercises involving trigonometric expressions and equations. They act as building blocks to deconstruct or reassemble expressions in a more manageable form.

Sine and Cosine Values

Knowing the sine and cosine values of standard angles is foundational in trigonometry. These values are often expressed as fractions or irrational numbers involving roots, which are derived from the unit circle.

For the standard angle \( \frac{5\pi}{4} \), located in the third quadrant of the unit circle, you need to remember:

• \( \cos \left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \): Since cosine is negative in the third quadrant.
• \( \sin \left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \): Sine is also negative in this quadrant.

With these values, you can substitute into trigonometric expressions to analyze or verify identities such as those using the cosine difference formula. Practicing with these values will enhance your confidence and precision in trigonometry.

Angle Formulas

Angle formulas, like the cosine of difference, are used to find the cosine or sine of an angle combination. In our case, the Cosine of Difference formula is crucial: \( \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \).

This formula helps you break down the cosine of a composite angle into a combination of sine and cosine of its individual angles. For instance, when applying this concept to \( \cos \left(x - \frac{5\pi}{4}\right) \), the formula allows you to express it based on the known values of \( \cos \left(\frac{5\pi}{4}\right) \) and \( \sin \left(\frac{5\pi}{4}\right) \).

Applying such formulas simplifies solving complex trigonometric equations and verifying identities by enabling piecewise computations. Understanding these formulas can expedite problem-solving processes and enhance accuracy.

Identity Verification

Identity verification in trigonometry involves proving or disproving that one side of the equation can be transformed into the other using known identities. This process leverages trigonometric properties and is integral in ensuring that mathematical expressions hold true under all valid conditions of the variables.

In our given exercise, we applied the cosine of difference formula to simplify \( \cos \left(x - \frac{5\pi}{4}\right) \) and compared it to \( -\frac{\sqrt{2}}{2} (\cos x + \sin x) \). Through step-by-step simplification, we determined that the expression simplified to \( -\frac{\sqrt{2}}{2}(\cos x - \sin x) \), which did not verify the proposed identity.

Issues in identity verification often arise due to errors in calculation or the inadequacy of the proposed identity. Successful verification strengthens your grasp of trigonometric relations, and when identities do not hold, analyzing discrepancies offers learning opportunities.