Problem 47 - Algebra and Trigonometry

Question

Find the exact value of the expression. $$ \sin \left(\cos ^{-1} \frac{1}{2}+\tan ^{-1} 1\right) $$

Step-by-Step Solution

Answer

The exact value is \( \frac{\sqrt{6} + \sqrt{2}}{4} \).

Step 1: Understand Inverse Trigonometric Functions

First, identify what the inverse trigonometric functions in the expression imply. Given that \( \cos^{-1}(\frac{1}{2}) = \theta \), this means that \( \cos(\theta) = \frac{1}{2} \). We know this scenario corresponds to \( \theta = \frac{\pi}{3} \) because the cosine of 60 degrees (or \( \pi/3 \) radians) is 1/2.For \( \tan^{-1}(1) = \phi \), this means that \( \tan(\phi) = 1 \). This corresponds to \( \phi = \frac{\pi}{4} \) because the tangent of 45 degrees (or \( \pi/4 \) radians) is 1.

Step 2: Find the Expression Inside Sine Function

We need to add the two angles obtained from the inverse functions: \[ \cos^{-1}\left(\frac{1}{2}\right) + \tan^{-1}(1) = \frac{\pi}{3} + \frac{\pi}{4}. \]

Step 3: Add the Angles

Add the two angles by finding a common denominator:\[\frac{\pi}{3} + \frac{\pi}{4} = \frac{4\pi}{12} + \frac{3\pi}{12} = \frac{7\pi}{12}.\]

Step 4: Use the Sine Addition Formula

Now, you want to find \( \sin \left( \frac{7\pi}{12} \right) \). To make this computation easier, use the sine addition formula:\[ \sin(a + b) = \sin a \cos b + \cos a \sin b, \]where here we have:\[ a = \frac{\pi}{3}, \quad b = \frac{\pi}{4}. \]

Step 5: Substitute Values into the Formula

Substitute the trigonometric values for \( \sin(a) \), \( \cos(a) \), \( \sin(b) \), and \( \cos(b) \):\[\sin \left( \frac{7\pi}{12} \right) = \left( \sin \frac{\pi}{3} \cdot \cos \frac{\pi}{4} \right) + \left( \cos \frac{\pi}{3} \cdot \sin \frac{\pi}{4} \right)\]\[ = \left( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \right) + \left( \frac{1}{2} \cdot \frac{\sqrt{2}}{2} \right). \]

Step 6: Simplify the Expression

Now, calculate and simplify:\[= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}. \]

Step 7: Conclusion

The exact value of the expression \( \sin \left( \cos^{-1} \frac{1}{2} + \tan^{-1} 1 \right) \) is \( \frac{\sqrt{6} + \sqrt{2}}{4} \).

Key Concepts

Sine Addition Formula

The sine addition formula is a fundamental tool in trigonometry. It is used to find the sine of a sum of two angles. This formula is particularly useful when the angles are not standard ones found on the unit circle. The formula is:

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

This equation lets you compute the sine of an angle formed by the addition of other angles. Remember, each trigonometric component, such as \( \sin \) or \( \cos \), relates back to right triangle definitions or the unit circle. To effectively use this formula, one must be able to determine the sine and cosine values of the individual angles \( a \) and \( b \).

Exact Value

Finding the exact value in trigonometric problems often involves expressions that can be reduced to specific, well-known angles. In this exercise, we needed to find the exact value of \( \sin \left(\cos^{-1} \frac{1}{2} + \tan^{-1} 1\right) \).
Using inverse trigonometric functions helped us identify these angles as \( \theta = \frac{\pi}{3} \) and \( \phi = \frac{\pi}{4} \), respectively.
When plugged into the sine addition formula, these familiar angles generated values that are calculable without approximation:

• \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \)
• \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
• \( \cos \frac{\pi}{3} = \frac{1}{2} \)
• \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)

Calculations with these values lead directly to an exact expression.

Trigonometric Identities

Trigonometric identities are mathematical equations that express relationships between trigonometric functions. They are pivotal in simplifying complex trigonometric problems and calculations. Other than the sine addition formula, identities like the Pythagorean identities or angle sum identities are useful:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Angle Sum Identity for tangent: \( \tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \)

These identities help find exact values and evaluate expressions like the one in the exercise efficiently. Understanding these identities is critical for tackling a variety of trigonometric problems.

Angle Addition

Angle addition is a fundamental concept in trigonometry, involving the sum of two or more angles. Here, we added the angles \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \) to determine \( \frac{7\pi}{12} \).
Using the sine addition formula subsequently helped evaluate the sine of this sum. Correctly finding the sum requires recognizing common denominators, as done in the problem to convert \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \) to \( \frac{4\pi}{12} \) and \( \frac{3\pi}{12} \).
Employing angle addition techniques allows us to express complex angles in terms of simpler, more familiar angles, making it easier to utilize established trigonometric values.