Problem 104
Question
Use the table feature of a graphing utility to demonstrate the identity for each value of \(\theta\) $$\sin (-\theta)=-\sin \theta$$ (a) \(\theta=250^{\circ}\) (b) \(\theta=\frac{\pi}{4}\)
Step-by-Step Solution
Verified Answer
The identity \(\sin (-\theta)=-\sin \theta\) has been verified for both angles using a graphing utility.
1Step 1: Verify the identity for \(\theta=250^{\circ}\)
First, make sure the calculator is in degree mode. Then, calculate \(\sin(-250^{\circ})\) and \(-\sin(250^{\circ})\) using the graphing utility. If both values are the same, then the identity is verified for \(\theta=250^{\circ}\).
2Step 2: Verify the identity for \(\theta=\frac{\pi}{4}\)
Switch the calculator to radian mode. Then calculate \(\sin(-\frac{\pi}{4})\) and \(-\sin(\frac{\pi}{4})\) using the graphing utility. If both values are the same, then the identity is verified for \(\theta=\frac{\pi}{4}\).
Key Concepts
Graphing UtilityDegrees and RadiansSine Function
Graphing Utility
A graphing utility is a powerful tool that provides a visual representation of mathematical equations and functions. These utilities are often used in math and science to analyze data and perform complex calculations quickly.
One of the most common uses of a graphing utility is to plot functions to see their behavior and check mathematical identities.
In our exercise, a graphing utility is employed to demonstrate the trigonometric identity:
\(\sin(-\theta) = -\sin(\theta)\).
One of the most common uses of a graphing utility is to plot functions to see their behavior and check mathematical identities.
In our exercise, a graphing utility is employed to demonstrate the trigonometric identity:
\(\sin(-\theta) = -\sin(\theta)\).
- It can quickly switch between degree and radian modes, which is essential for checking angles in different measurements.
- Graphing functions visually, confirming algebraic identities visually.
- Also calculating values for specific inputs to confirm mathematical theories and formulas.
Degrees and Radians
Understanding degrees and radians is crucial in trigonometry. Both are units of angle measurement, and each has its own application settings.
Degrees are perhaps more familiar as we use them in everyday life. A full circle is commonly understood to have 360 degrees.
On the other hand, radians provide a more natural mathematical measure. A full circle is equal to \(2\pi\) radians.
In trigonometric functions, both measure systems can be used. Yet, often radians are preferred for their simplicity in mathematical equations due to their relation with the radius and circumference of a circle.
When solving problems involving trigonometric identities, ensure you're using the right unit, as it affects your calculations.
If you mix up degrees and radians in your calculations, you can arrive at incorrect results. So, always check the mode your calculator is set to.
Degrees are perhaps more familiar as we use them in everyday life. A full circle is commonly understood to have 360 degrees.
On the other hand, radians provide a more natural mathematical measure. A full circle is equal to \(2\pi\) radians.
In trigonometric functions, both measure systems can be used. Yet, often radians are preferred for their simplicity in mathematical equations due to their relation with the radius and circumference of a circle.
When solving problems involving trigonometric identities, ensure you're using the right unit, as it affects your calculations.
If you mix up degrees and radians in your calculations, you can arrive at incorrect results. So, always check the mode your calculator is set to.
- Degrees: 360° in a full circle.
- Radians: \(2\pi\) radians in a full circle.
- Switching modes changes how the input angle is interpreted.
Sine Function
The sine function is a fundamental concept in trigonometry. It helps to describe the relationship between the angle and the ratio of two sides of a right triangle.
Given an angle \(\theta\),
\(\sin(\theta) = \text{opposite side} / \text{hypotenuse}\).
In our exercise, the verification of the identity \(\sin(-\theta) = -\sin(\theta)\) uses the graphing utility to test negative and positive angles. By computing these values with the utility, you can see the symmetry and periodicity properties in action, strengthening understanding of the sine function's behavior.
Given an angle \(\theta\),
\(\sin(\theta) = \text{opposite side} / \text{hypotenuse}\).
Properties of the Sine Function
The sine function has some distinctive properties:- Periodicity: It repeats its values in regular intervals. The period is \(2\pi\) for radians and 360° for degrees.
- Range: The values y of \(\sin(\theta)\) are bound between -1 and 1.
- Symmetry: It is an odd function, meaning \(\sin(-\theta) = -\sin(\theta)\).
In our exercise, the verification of the identity \(\sin(-\theta) = -\sin(\theta)\) uses the graphing utility to test negative and positive angles. By computing these values with the utility, you can see the symmetry and periodicity properties in action, strengthening understanding of the sine function's behavior.
Other exercises in this chapter
Problem 103
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