Problem 122 - Precalculus with Limits

Question

In Exercises 119 - 122, use a calculator to demonstrate the identity for each value of \( \theta \). \( \sin(-\theta) = - \sin \theta \) (a) \( \theta = 250^circ \) (b) \( \theta = \dfrac{1}{2} \)

Step-by-Step Solution

Answer

The identity \( \sin(-\theta) = - \sin \theta \) holds for \( \theta = 250^circ \) and \( \theta = \dfrac{1}{2} \).

Step 1: Calculate \( \sin(-\theta) \) for \( \theta = 250^circ \)

Use your calculator to calculate the value of \( \sin(-250^circ) \)

Step 2: Calculate \( -\sin (\theta) \) for \( \theta = 250^circ \)

Now, find the value of \( -\sin (250^circ) \) using your calculator. You should get the same value as the one in the previous step.

Step 3: Compare the values

Both steps 1 and 2 should give the same results, showing that \( \sin(-\theta) = - \sin \theta \) for \( \theta = 250^circ \).

Step 4: Calculate \( \sin(-\theta) \) for \( \theta = \dfrac{1}{2} \)

Use your calculator to calculate the value of \( \sin(-\dfrac{1}{2}) \). It's important to note that your calculator must be in radian mode because the value of \( \theta \) is given in radians.

Step 5: Calculate \( -\sin (\theta) \) for \( \theta = \dfrac{1}{2} \)

Now, find the value of \( -\sin (\dfrac{1}{2}) \) using your calculator. You should get the same value as the one in the previous step.

Step 6: Compare the values

Both steps 4 and 5 should give the same results, showing that \( \sin(-\theta) = - \sin \theta \) for \( \theta = \dfrac{1}{2} \).

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function, often abbreviated as "sin." It relates a given angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. Mathematically, it is expressed as:\[\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}\]For any angle \(\theta\), the sine function takes on values between -1 and 1, inclusive. It is important to learn that the sine function is periodic and repeats every \(360^\circ\) or \(2\pi\) radians. This periodicity means that for any angle \(\theta\), you can add or subtract \(360^\circ\) or \(2\pi\) to get an equivalent angle with the same sine value.

Another essential identity involving sine is \(\sin(-\theta) = -\sin \theta\). This tells us that the sine function is **odd**, meaning that flipping an angle across the origin results in a change in the sign of its sine value. By verifying this identity using specific angles, such as in the exercise, we reinforce our understanding of this concept.

Angle Measurement

Understanding angle measurement is crucial when working with trigonometric functions. Angles can be measured in degrees or radians. Here’s a quick breakdown:

• **Degrees:** A full circle is \(360^\circ\). Each degree is \(1/360\) of a complete circle.
• **Radians:** A full circle is \(2\pi\) radians. Each radian is the angle formed when the arc length is equal to the radius of the circle.

Degrees are more commonly used in daily life, like in navigation and geometry. However, radians are the preferred measurement in many mathematical contexts, especially calculus. This is because radian measure directly relates arc length to the radius of a circle.

In the context of the exercise, notice how the angles are given in different units. When working with angles in trigonometric identities, ensure that you are consistent with angle measurement, especially when using calculators.

Calculator Usage

When demonstrating trigonometric identities using a calculator, it's essential to perform the correct procedures to avoid errors. Here are some tips:

• **Check the Mode:** Make sure the calculator is in the correct mode (degree or radian) that corresponds to your angle measurement.
• **Use Functions Correctly:** If calculating \(\sin(-\theta)\), input \(-\theta\) accurately. If you need \(-\sin(\theta)\), compute \(\sin(\theta)\) first, then apply the negative sign.
• **Verify Accuracy:** Compare your results from both sides of the identity to confirm correctness.

Always ensure that your calculations are consistent with mathematical rules and calculator settings. Practicing these steps can develop your ability to accurately verify trigonometric identities.

Radian Mode

Knowing what radian mode is and when to use it is pivotal in trigonometry. Radian mode on your calculator ensures that it interprets angle inputs as radians rather than degrees. This is critical because radians are the standard unit in higher mathematics.In the exercise, when \(\theta = \frac{1}{2}\), this is implicitly in radian measure. Therefore, your calculator should be in radian mode to correctly interpret this value. To switch between modes, you typically press a mode button, then select the radian or degree option. This tells your calculator exactly how to handle inputs, avoiding any unintended calculations.

Without proper knowledge of radian mode, calculations involving radians might yield incorrect results. Always set your calculator mode corresponding to how the angle is measured to ensure accuracy in calculating trigonometric functions.