Problem 70 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Give exact values for \(\sin 75^{\circ}\) and \(\cos 15^{\circ}\)

Step-by-Step Solution

Answer

\( \sin 75^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \) and \( \cos 15^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \)

Step 1: Use Angle Sum and Difference Formulas

To find the exact values for \( \sin 75^{\circ}\) and \( \cos 15^{\circ}\), we can use the angle sum and difference identities. Recall that: \[ \sin(A + B) = \sin A \cos B + \cos A \sin B \] and \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \].

Step 2: Break Down the Angles

\(75^{\circ}\) can be written as \(45^{\circ} + 30^{\circ}\) and \(15^{\circ}\) can be written as \(45^{\circ} - 30^{\circ}\). This matches the form needed for the sum and difference identities.

Step 3: Apply the Angle Sum Formula for \(\sin 75^{\circ}\)

Using the identity \( \sin(45^{\circ} + 30^{\circ}) = \sin 45^{\circ} \cos 30^{\circ} + \cos 45^{\circ} \sin 30^{\circ} \), substitute the known values: \[ \sin 45^{\circ} = \frac{\sqrt{2}}{2}, \cos 30^{\circ} = \frac{\sqrt{3}}{2}, \cos 45^{\circ} = \frac{\sqrt{2}}{2}, \sin 30^{\circ} = \frac{1}{2} \]. Therefore: \[ \sin 75^{\circ} = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \].

Step 4: Apply the Angle Difference Formula for \(\cos 15^{\circ}\)

Using the identity \( \cos(45^{\circ} - 30^{\circ}) = \cos 45^{\circ} \cos 30^{\circ} + \sin 45^{\circ} \sin 30^{\circ} \), substitute the known values: \[ \cos 45^{\circ} = \frac{\sqrt{2}}{2}, \cos 30^{\circ} = \frac{\sqrt{3}}{2}, \sin 45^{\circ} = \frac{\sqrt{2}}{2}, \sin 30^{\circ} = \frac{1}{2} \]. Therefore: \[ \cos 15^{\circ} = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \].

Key Concepts

Angle Sum and Difference Formulas

To solve trigonometric problems involving angles, the angle sum and difference formulas come in handy. These identities allow us to break down complex angles into sums or differences of simpler, known angles.
For example, for any angles A and B:

• The angle sum formula for sine is: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \).
• The angle difference formula for cosine is: \( \cos(A - B) = \cos A \cos B + \sin A \sin B \).

Using these, we can compute exact values for trigonometric functions for non-standard angles by rewriting them in terms of standard angles like 30°, 45°, and 60°.

Exact Values of Trigonometric Functions

Knowing the exact values of trigonometric functions for specific angles is fundamental. These values are typically based on right triangles and the unit circle.
For instance, for special angles:

• \( \sin 45\^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \cos 45\^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin 30\^{\circ} = \frac{1}{2} \)
• \( \cos 30\^{\circ} = \frac{\sqrt{3}}{2} \)

Memorizing these values helps in quickly solving trigonometric problems involving these angles. Use these values in conjunction with angle sum and difference formulas to find exact trigonometric values for other angles.

Trigonometric Evaluation

Evaluating trigonometric functions for angles not directly found on the unit circle typically involves the use of identities and known angle values.
Consider evaluating \( \sin 75\^{\circ} \) and \( \cos 15\^{\circ} \):

• First, rewrite the given angles in forms suitable for the angle sum and difference formulas: \( 75\^{\circ} = 45\^{\circ} + 30\^{\circ} \) and \( 15\^{\circ} = 45\^{\circ} - 30\^{\circ} \).
• Apply the angle sum formula for sine: \( \sin(75\^{\circ}) = \sin(45\^{\circ} + 30\^{\circ}) = \sin 45\^{\circ} \cos 30\^{\circ} + \cos 45\^{\circ} \sin 30\^{\circ} = \frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4} \).
• Apply the angle difference formula for cosine: \( \cos(15\^{\circ}) = \cos(45\^{\circ} - 30\^{\circ}) = \cos 45\^{\circ} \cos 30\^{\circ} + \sin 45\^{\circ} \sin 30\^{\circ} = \frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4} \).

Using these methods helps to not only find the exact values but also to understand and apply trigonometric identities more effectively.