Problem 53 - Algebra and Trigonometry

Question

Use a sketch to find the exact value of each expression. $$ \sin \left(\cos ^{-1} \frac{\sqrt{2}}{2}\right) $$

Step-by-Step Solution

Answer

The exact value of \( \sin \left(\cos ^{-1} \frac{\sqrt{2}}{2}\right) \) is \( \frac{\sqrt{2}}{2} \).

Step 1: Understanding Inverse Cosine Function

The expression \( \cos ^{-1} \frac{\sqrt{2}}{2} \) is nothing but the angle whose cosine is \( \frac{\sqrt{2}}{2} \). From basic trigonometry, each of \(\cos \frac{\pi}{4} \) and \( \cos \frac{-\pi}{4} \) equals \( \frac{\sqrt{2}}{2} \). But, the range of the inverse cosine function being \( [0, \pi] \), it would return \( \frac{\pi}{4} \). So, \( \cos ^{-1} \frac{\sqrt{2}}{2} = \frac{\pi}{4} \).

Step 2: Calculating the Sine of the Angle

Now, we need to find the sine of the angle \( \frac{\pi}{4} \). From a standard right triangle or the unit circle, we know that \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 3: Substitution

Finally, substituting \( \frac{\pi}{4} \) back into the original expression we get \( \sin \left(\cos ^{-1} \frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{2} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in trigonometry that allow us to find the angles when the function value is given. In our problem, we have the expression \( \cos^{-1} \frac{\sqrt{2}}{2} \). This represents the angle whose cosine value is \( \frac{\sqrt{2}}{2} \). Trigonometric functions, like cosine, repeatedly output the same values for different angles. The job of the inverse cosine function, however, is to give us only one angle in the range \( [0, \pi] \) where the cosine is exactly \( \frac{\sqrt{2}}{2} \).
For the cosine inverse function:

• The range is \( [0, \pi] \)
• The result is always a principal angle

In this particular problem, \( \cos^{-1} \frac{\sqrt{2}}{2} = \frac{\pi}{4} \) because \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \). It's crucial to understand these functions because they allow us to handle triangles and rotations in advanced mathematics.

Unit Circle

The unit circle is an essential concept in trigonometry, serving as a visual guide to understand how trigonometric functions behave. A unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. The unit circle helps in understanding:

• The relationships between angles and coordinates (x,y)
• How cosine and sine relate to x and y coordinates

At the angle \( \frac{\pi}{4} \), both x and y coordinates are equal to \( \frac{\sqrt{2}}{2} \), since \( \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \). This is because the unit circle's radius is 1, allowing these trigonometric relationships to be directly visualized from the circle. By recognizing points on the unit circle, we can easily determine the sine and cosine of angles that correspond to these points.

Right Triangle Trigonometry

Right triangle trigonometry is a fundamental branch of trigonometry focusing on the relationships between the angles and sides of right-angled triangles. In this system:

• Sine of an angle is the ratio of the opposite side to the hypotenuse.
• Cosine of an angle is the ratio of the adjacent side to the hypotenuse.

When solving for \( \sin \frac{\pi}{4} \), we visualize the right triangle with a \( \frac{\pi}{4} \) angle (or 45 degrees). In such a triangle, both non-hypotenuse sides are of equal length because the triangle is isosceles. The ratio of either side to the hypotenuse, which is \( \sqrt{2} \), makes \( \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \).
Thus, right triangle trigonometry helps us compute exact trigonometric values, which simplifies solving problems that involve inverse functions and the unit circle.