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Chapter 20: Problem 33

Use a calculator to evaluate the given expressions. $$\tan \left[\cos ^{-1}(-0.6281)\right]$$

Short Answer

Expert verified
\( \tan \left[ \cos^{-1}(-0.6281) \right] \approx -0.7373 \)

Step by step solution

01

Understand the Inverse Cosine Function

The expression inside the tangent function is \( \cos^{-1}(-0.6281) \). This function, \( \cos^{-1}(x) \), finds the angle \( \theta \) such that \( \cos(\theta) = x \). So, we're looking for the angle \( \theta \) whose cosine is \(-0.6281\). This angle is given in radians.
02

Calculate \( \theta = \cos^{-1}(-0.6281) \)

Using a calculator set to radian mode, input \( \cos^{-1}(-0.6281) \). The result is approximately \( 2.2522 \) radians. This angle \( \theta \) is in Quadrant II because the cosine is negative.
03

Use the Angle to Evaluate the Tangent

Now, with \( \theta = 2.2522 \) radians from Step 2, we need to find \( \tan(\theta) \). Use a calculator to find the tangent of this angle: \( \tan(2.2522) \approx -0.7373 \).
04

Check Quadrant and Sign

Since \( \theta \) is in Quadrant II (cosine is negative while sine is positive), the tangent in Quadrant II is negative. Our calculated \( \tan(2.2522) \approx -0.7373 \) confirms this since it is negative, as expected.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Cosine
The inverse cosine function, denoted as \( \cos^{-1}(x) \), is an essential concept in trigonometry. It helps you find the angle whose cosine value is a known number. Imagine knowing how wide a shadow is, but needing to figure out the angle of the sun creating it. Similarly, with \( \cos^{-1}(-0.6281) \), we're looking for the specific angle \( \theta \) such that \( \cos(\theta) = -0.6281 \).
  • This function only has outputs in the range of \( [0, \pi] \) radians, restricted to angles where the cosine value is meaningful.
  • The function is particularly useful when dealing with negative cosine values, implying the angle lies in the second quadrant.
Knowing how to find angles from cosine values is foundational in solving more complex trigonometric problems.
Angle Measurement in Radians
Understanding radians is crucial for working with angles in trigonometry. A radian is another way to measure angles, different from degrees. One full circle is \(2\pi\) radians, or equivalently, 360 degrees. This natural measure of angles is widely used because it simplifies many mathematical formulas, especially in calculus.
  • An angle measured in radians is based on the radius of a circle, giving it a connection to the circle鈥檚 arc length.
  • Often, calculators set to radian mode ensure results are consistent when dealing with trigonometric functions.
When we calculated \( \cos^{-1}(-0.6281) \), the angle \( \theta \) came out to approximately \(2.2522\) radians. Being in radians helps in directly plugging it into other trigonometric functions like tangent.
Tangent Function
The tangent function, denoted \( \tan(\theta) \), relates the sine and cosine of an angle. It is defined as the ratio \( \frac{\sin(\theta)}{\cos(\theta)} \). In our context, after finding the angle using inverse cosine, we determine its tangent to complete the exercise.
  • Tangent is particularly interesting because its value can vary widely and includes negative numbers too.
  • In the second quadrant, as in our example, tangent will always be negative because sine is positive while cosine is negative.
By plugging \( \theta = 2.2522 \) into the tangent function, we find that \( \tan(2.2522) \approx -0.7373 \), matching our expectation for the second quadrant.

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Most popular questions from this chapter

Prove that the given expressions are equal. In Exercise \(57,\) use the relation for \(\sin (\alpha+\beta)\) and show that the sine of the sum of the angles on the left equals the sine of the angle on the right. In Exercise \(58,\) use the relation for \(\tan (\alpha+\beta).\) $$\sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{5}{13}=\sin ^{-1} \frac{56}{65}$$

Evaluate the given expressions. $$\sin ^{-1} 0.5+\cos ^{-1} 0.5$$

Find an algebraic expression for each of the given expressions. $$\sin \left(\sec ^{-1} \frac{x}{4}\right)$$

Use the given substitutions to show that the given equations are valid. In each, \(0<\theta<\pi / 2\). $$\text { If } x=3 \sin \theta, \text { show that } \sqrt{9-x^{2}}=3 \cos \theta$$.

Solve the given equations graphically. An equation used in astronomy is \(\theta-e \sin \theta=M .\) Solve for \(\theta\) for \(e=0.25\) and \(M=0.75\).
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