Problem 13 - Precalculus with Calculus Previews

Question

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right) $$

Step-by-Step Solution

Answer

The exact value is \(-\frac{\pi}{4}\).

Step 1: Understand the Function

The given expression is written as \( \sin^{-1}\left(-\frac{\sqrt{2}}{2}\right) \). The expression \( \sin^{-1}(x) \), or arcsin, is the inverse function of \( \sin(x) \), which finds the angle whose sine is \( x \).

Step 2: Recall Sine Values

The value \(-\frac{\sqrt{2}}{2}\) is a known value for the sine function for angles in the unit circle. Specifically, it corresponds to the angles where sine is negative in the reference angles of \( \frac{\pi}{4} \) (or 45 degrees).

Step 3: Determine the Quadrants

Since \(-\frac{\sqrt{2}}{2}\) is negative, we look for angles in the third (quadrant III) and fourth (quadrant IV) quadrants. However, arcsin, limited to the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\), only includes the fourth quadrant.

Step 4: Find the Exact Angle

The reference angle whose sine is \( \frac{\sqrt{2}}{2} \) is \( \frac{\pi}{4} \). In the fourth quadrant, the angle corresponding to this negative sine is \( -\frac{\pi}{4} \) since we take the negative of the reference angle.

Step 5: Verify the Solution

Check that \( \sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \). This confirms the correctness because, in the unit circle, \(-\frac{\pi}{4}\) does correspond to \(-\frac{\sqrt{2}}{2}\) for \( \sin(x) \).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. The significance of the unit circle in trigonometry lies in its ability to represent angles and their corresponding trigonometric values very simply. Any point on the unit circle can be represented as \((\cos(\theta), \sin(\theta))\), where \(\theta\) is the angle formed with the positive x-axis. This means for any given angle, you can directly find its cosine and sine values by identifying the respective x and y coordinates of the point on the unit circle.

• Angles are usually measured in radians when using the unit circle.
• The full circle is \(2\pi\) radians around.

When solving problems with inverse sine or other trigonometric functions, the unit circle helps identify the possible angles that can correspond to specific values. For example, if you are dealing with an inverse sine function, you are finding which angle corresponds to a given sine value. Using the unit circle effectively helps in visualizing and understanding which quadrant an angle might be in, significantly aiding in solving trigonometric expressions without a calculator.

Sine Function

The sine function is a key element of trigonometry and defines a relationship between an angle and the length of the opposite side of a right triangle divided by the length of the hypotenuse. In the context of the unit circle, it specifically refers to the y-coordinate of a point on the circle. Mathematically, the sine function is defined as follows:\[\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}\]On the unit circle, since the hypotenuse is always 1, \(\sin(\theta)\) is simply the y-coordinate. This makes evaluating angles on the unit circle straightforward, as you only need to refer to the "height" (y-value) of the point for any given \(\theta\).

• The sine function is periodic with a period of \(2\pi\), indicating that its values repeat every \(2\pi\) radians.
• Its range is between \([-1, 1]\), meaning it never takes on values outside this interval.

Inverse sine functions, such as \(\sin^{-1}(x)\), allow you to find angles when given a sine value. They typically return values within the principal range \([\frac{-\pi}{2}, \frac{\pi}{2}]\), focusings mainly on the first and fourth quadrants to provide single, clear answers for every sine value within its range.

Trigonometric Identities

Trigonometric identities are formulas that express relationships involving the trigonometric functions that are true for every value of the variables involved. These identities are crucial in simplifying expressions and solving trigonometric equations.Some key identities include:

• Pythagorean Identities: Relate \(\sin\), \(\cos\), and \(\tan\) functions, most famously \(\sin^2(\theta) + \cos^2(\theta) = 1\).
• Angle Sum and Difference Identities: Allow calculation of the sine, cosine, and tangent of sum or difference of two angles.
• Reciprocal Identities: Concern relationships between \(\sin\),\(\cos\), \(\tan\) and their reciprocals, \(\csc\), \(\sec\), and \(\cot\).

In solving the given exercise, understanding that \(\sin^{-1}(-\frac{\sqrt{2}}{2})\) requires recognizing it involves a specific angle where sine outputs a negative value. Here, the knowledge of identities helps in confirming angle positions — like knowing negative values of sine inverts the reference angle within specified quadrants. Such identities also aid in verifying your results, ensuring that the calculated angle indeed satisfies the original trigonometric equation.