Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the value of the expression without using a calculator. $$\sin \left(\sin ^{-1} 1+\cos ^{-1} 1\right)$$

Step-by-Step Solution

Answer

The value of the expression \( \sin(\sin^{-1} 1+\cos^{-1} 1) \) is 1.

Step 1: Find the Values of Inverse Trigonometric Functions

We begin by finding the values of \( \sin^{-1} 1 \) and \( \cos^{-1} 1 \). \( \sin^{-1} 1 \) gives the angle whose sine is 1, and \( \cos^{-1} 1 \) gives the angle whose cosine is 1. In the unit circle, the angle whose sine is 1 is \( \frac{\pi}{2} \) (90 degrees) and the angle whose cosine is 1 is 0. So, we have \( \sin^{-1} 1 = \frac{\pi}{2} \) and \( \cos^{-1} 1 = 0 \).

Step 2: Substitute the Values Into the Expression

Next, we substitute these values back into the expression. We have \( \sin (\sin^{-1} 1+\cos^{-1} 1) = \sin (\frac{\pi}{2} + 0) \).

Step 3: Simplify the Expression

The expression simplifies to \( \sin(\frac{\pi}{2}) \), which is 1. So, the value of the expression \( \sin(\sin^{-1} 1+\cos^{-1} 1) \) is 1.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential for finding angles when you know a ratio of sides in right triangles. Specifically, these functions "undo" the trigonometric functions like sine, cosine, and tangent.

For instance:

• \( \sin^{-1}(x) \) gives the angle whose sine is \( x \).
• \( \cos^{-1}(x) \) gives the angle whose cosine is \( x \).
• \( \tan^{-1}(x) \) gives the angle whose tangent is \( x \).

These functions play a significant role in trig equations and graphs because they help us find specific angles. When we compute \( \sin^{-1}(1) \), we are seeking the angle \( \frac{\pi}{2} \) (90 degrees), as this is the angle on the unit circle where the sine equals 1. Similarly, \( \cos^{-1}(1) \) points us to the angle 0, since cosine equals 1 at this point on the unit circle.

Unit Circle

The unit circle is a powerful tool in trigonometry, simplifying complex problems. It is a circle with a radius of 1 centered at the origin

• The circle's unique properties allow you to easily find sine and cosine values for various angles.
• Each angle corresponds to a unique point on the circle, where the x-coordinate gives the cosine value and the y-coordinate gives the sine value.

For example, on the unit circle:

• The angle \(0\) is at point \((1, 0)\), where the cosine of the angle is 1 and the sine of the angle is 0.
• The angle \( \frac{\pi}{2} \) is at point \((0, 1)\), where the cosine is 0 and the sine is 1.

These points aid in quickly computing trigonometric values without needing a calculator. Hence, combining angles like \( \frac{\pi}{2} \) and 0 enables us to determine expressions like \( \sin(\frac{\pi}{2} + 0) \).

Trigonometric Identities

Trigonometric identities are equations that hold true for all angles. They are fundamental tools for simplifying trigonometric expressions and solving equations. Key identities include:

• The Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Angle Addition Formulas, such as \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \)
• Double Angle Formulas, like \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)

In the given problem, although indirect, our understanding of the identity \( \sin(\sin^{-1}(1) + \cos^{-1}(1)) \) leans on the realization that these angles conveniently align to form known outputs of the trigonometric functions. Recognizing these identities and how angles on the unit circle map to specific trigonometric values makes seemingly intricate problems much more approachable.