Question

Without using a calculator, evaluate, if possible, the following expressions. $$\sin ^{-1}(-1)$$

Short answer

Answer: The angle with a sine value of -1 is $$270^{\circ}$$ or $$\frac{3}{2}\pi$$ radians.

Step-by-step solution

Identify the correct quadrant

To find an angle with a negative sine, we need to consider the quadrants where the sine is negative. Recall that sine represents the y-coordinate of a point on the unit circle. Therefore, the sine is negative in quadrants III and IV.

Find the angle

We need to find the angle θ whose sine is -1, and we know thatθ lies in either quadrant III or IV. Recall that sin(θ) = -1 only occurs at a specific point on the unit circle, and that point is where the y-coordinate is -1. The point on the unit circle where the y-coordinate is -1 is (0, -1). This point corresponds to an angle θ in the standard position (with the initial side on the positive x-axis) of $$\theta = 270^{\circ}$$ or $$\theta = \frac{3}{2}\pi$$ radians.

Write the final answer

Now that we've found the angle whose sine is -1, we can write the final answer. The given expression, $$\sin ^{-1}(-1)$$, evaluates to: $$\sin ^{-1}(-1) = 270^{\circ}$$ or $$\sin^{-1}(-1) = \frac{3}{2}\pi$$ radians.

Key concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one and is centered at the origin of a coordinate plane. In the unit circle, every point represents an angle measured from the positive x-axis. Understanding the unit circle allows us to determine the trigonometric values of angles.

Here are some key points regarding the unit circle:

• The circumference of the circle is divided into 360 degrees or \(2\pi\) radians.
• The coordinates of points on the unit circle correspond to the cosine and sine of the angle, \((\cos\theta, \sin\theta)\).
• For example, the point (0, -1) on the unit circle corresponds to an angle of \(270^{\circ}\) or \(\frac{3}{2}\pi\) radians.

Each position on the unit circle is unique and helps to find the sine, cosine, and tangent of the angle. These values are crucial in solving trigonometric equations and understanding advanced trigonometric functions.

Trigonometric Quadrants

The coordinate plane is divided into four quadrants, each affecting the sign of trigonometric functions differently. These quadrants help us understand where functions like sine, cosine, and tangent are positive or negative:

• Quadrant I (0° to 90° or 0 to \(\frac{\pi}{2}\) radians): All trigonometric functions (sine, cosine, and tangent) are positive.
• Quadrant II (90° to 180° or \(\frac{\pi}{2}\) to \(\pi\) radians): Sine is positive, while cosine and tangent are negative.
• Quadrant III (180° to 270° or \(\pi\) to \(\frac{3\pi}{2}\) radians): Tangent is positive, while sine and cosine are negative.
• Quadrant IV (270° to 360° or \(\frac{3\pi}{2}\) to \(2\pi\) radians): Cosine is positive, while sine and tangent are negative.

In the original exercise, since the sine of a negative angle is involved, we know that the angle is either in Quadrant III or IV. This fact helped us pinpoint the solution, reinforcing the importance of understanding trigonometric quadrants.

Radians and Degrees

Radians and degrees are two ways of measuring angles. It's important to be familiar with both, as they are commonly used in trigonometry and other branches of mathematics. Here's a simple breakdown of these two units:

• Degrees: A full circle is 360 degrees. Common reference angles include 90°, 180°, and 270°.
• Radians: A full circle is \(2\pi\) radians. Key angles are \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\).
• The conversion between these units is vital: \(360^{\circ} = 2\pi\) radians means that \(1^{\circ} = \frac{\pi}{180}\) radians.

In the solution provided, the angle \(270^{\circ}\) was also given as \(\frac{3}{2}\pi\) radians. This dual representation highlights the importance of being comfortable with both radians and degrees, allowing for smooth transitions depending on the context of the problem.