Problem 83 - Precalculus

Question

Use a calculator to approximate \(\sin 423^{\circ} .\) What do you expect \(\sin \left(-423^{\circ}\right)\) to be? Verify your answer with a calculator.

Step-by-Step Solution

Answer

\(sin 423° \approx 0.8910\) and \(sin (-423°) \approx -0.8910\).

Step 1: Convert 423° to standard position

To find the sine of an angle using a calculator, first convert the angle to less than 360°, which is the standard position. Subtract 360° from 423°: \(423° - 360° = 63°\). Now, we can use 63° to find the sine because \(423°\) is coterminal to \(63°\).

Step 2: Use the calculator to find \( sin 63°\)

Enter \(63°\) into the calculator and use the sine function to find its value. The calculator gives \(sin 63° \approx 0.8910\). Therefore, \(sin 423° \approx 0.8910\).

Step 3: Determining \( sin (-423°)\)

Since sine is an odd function, \(sin (- heta) = -sin ( heta)\). Therefore, \(sin (-423°) = -sin 423°\). Based on our earlier calculation, this means \(sin (-423°) = -0.8910\).

Step 4: Verify using a calculator

Set your calculator to degree mode and enter \(-423°\). Use the sine function to find its value: \(\sin (-423°) \approx -0.8910\). This confirms the theoretical application that \(\sin (-423°)\) is the negative of \(\sin 423°\).

Key Concepts

Sine Function

The sine function is one of the key trigonometric functions in mathematics. It is often abbreviated as "sin" and is crucial in understanding wave patterns, oscillations, and circular motions. The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

• The range of the sine function is from -1 to 1.
• This function is periodic with a period of 360° (or 2\(\pi\) radians).

The value of the sine function fluctuates between -1 and 1 as the angle varies, producing a smooth wave-like pattern. For example, \(\sin 0° = 0\), \(\sin 90° = 1\), and \(\sin 180° = 0\). In our example of \(\sin 423°\), since 423° exceeds 360°, we use the concept of coterminality to "wrap around" and find the equivalent angle within one full cycle, which in this case is 63°.

Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but may differ by multiples of a full circle (360°). This means you can add or subtract 360° from an angle to find another angle that lands in the same position on the unit circle.

• Two angles are coterminal if the difference between them is a multiple of 360°.
• They have the same sine, cosine, and tangent values because they end up in the same location on the unit circle.

In the problem given, we found the sine of 423° by recognizing that it is coterminal with 63°, because 423° - 360° = 63°. This adjustment, or "modulus" operation, simplifies calculations using trigonometric functions, allowing us to work with smaller and more manageable reference angles.

Odd Functions

An odd function has a specific symmetry when graphed, where \(f(-x) = -f(x)\). The sine function is a classic example of an odd function. This property is extremely useful in trigonometry and helps us understand transformations of angles.

• For the sine function, \(\sin(-\theta) = -\sin(\theta)\).
• This implies that the graph of the sine function is symmetric about the origin.

Understanding odd functions aids in predicting the behavior of trigonometric values for negative angles easily. In the step-by-step exercise, the property of odd functions allowed us to conclude that \(\sin(-423°) = -\sin(423°)\), which simplifies our calculation to finding the negative of the already calculated sine value of 423°.