Problem 47 - Precalculus Mathematics for Calculus

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: Identify the Known Values

We need the values of \( \cos^{-1} \frac{1}{2} \) and \( \tan^{-1} 1 \). These correspond to angles whose cosine and tangent are those values.\( \cos^{-1} \frac{1}{2} \) is \( \frac{\pi}{3} \) and \( \tan^{-1} 1 \) is \( \frac{\pi}{4} \).

Step 2: Express the Sum of Angles

The expression \( \sin \left(\cos^{-1} \frac{1}{2} + \tan^{-1} 1\right)\) becomes \( \sin \left(\frac{\pi}{3} + \frac{\pi}{4}\right)\).

Step 3: Use the Sine Addition Formula

According to the sine addition formula, \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). Use this to calculate \( \sin \left(\frac{\pi}{3} + \frac{\pi}{4}\right)\).

Step 4: Substitute Known Values

Let's substitute \( a = \frac{\pi}{3} \) and \( b = \frac{\pi}{4} \) into the formula: \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), \( \cos \frac{\pi}{3} = \frac{1}{2} \), \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 5: Calculate Using the Values

Substitute into the formula: \( \sin \left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2} \).

Step 6: Simplify the Expression

Calculate: \( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4} \) and \( \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} \). Add them to get \( \frac{\sqrt{6} + \sqrt{2}}{4} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles given specific trigonometric values. They are the reverse operations of the standard trigonometric functions and are used to determine the angles that correspond to a particular trigonometric ratio. The function \( \cos^{-1} \) or the inverse cosine function, assigns the angle whose cosine is \( \frac{1}{2} \). In our problem, \( \cos^{-1} \frac{1}{2} = \frac{\pi}{3} \) because \( \cos (\frac{\pi}{3}) = \frac{1}{2} \).
The \( \tan^{-1} \) or inverse tangent function provides the angle whose tangent is 1. Here, \( \tan^{-1} 1 = \frac{\pi}{4} \) because the tangent of \( \frac{\pi}{4} \) is 1.
Knowing these specific values from trigonometric identities is key in many such problems, as these reference angles frequently appear in exercises.

Sine Addition Formula

The sine addition formula is a powerful tool in trigonometry. This identity is used when you need to find the sine of the sum of two angles. Mathematically, it states that:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
This equation helps break down the calculation into more manageable parts by using known values of the sine and cosine of the individual angles.
In our problem, we applied the formula to \( a = \frac{\pi}{3} \) and \( b = \frac{\pi}{4} \). By substituting the known values of these angles, the formula lets us express \( \sin(\frac{\pi}{3} + \frac{\pi}{4}) \) in terms of simpler trigonometric values, which we then calculate to find the solution.

Angle Sum Identities

Angle sum identities are essential in combining or resolving trigonometric expressions involving sums and differences of angles. These identities include the sine, cosine, and tangent angle sum formulas. In trigonometry, such identities help simplify expressions that involve the addition or subtraction of angles.
They are very useful when exact values are needed, as they allow computations using known angles and their sine or cosine values. In this particular problem, the angle sum identities allow us to take complicated expressions presented as sums of inverse trig functions and evaluate their trigonometric functions more simply.
Using these identities, especially the sine addition formula, enables precise calculations for problems like \( \sin(\cos^{-1} \frac{1}{2} + \tan^{-1} 1) \) in our step-by-step solution.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the specific, often memorized values that come from well-known angles in the unit circle, particularly \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3} \), and \( \frac{\pi}{2} \).
These angles correspond to either simple fractions or square roots that make calculations straightforward.

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

Understanding these values enables you to solve expressions involving these fundamental angles quickly. They are the building blocks for solving many trigonometric problems, including our exercise where we used them to work out \( \sin(\frac{\pi}{3} + \frac{\pi}{4}) \) with the sine addition formula.