Problem 15 Use a calculator to find sine, c... [FREE SOLUTION]

91影视

Precalculus: An Investigation of Functions

David Lippman

$Math Studyset 91影视 Explanations$ Math

2017 Edition

Chapter 5: Problem 15

Use a calculator to find sine, cosine, and tangent of the following values: a. 0.15 b. 4 c. $70^{\circ}$ d. $283^{\circ}$

Short Answer

Expert verified

For 0.15: sine 鈮� 0.1494, cosine 鈮� 0.9888, tangent 鈮� 0.1511. For 4: sine 鈮� -0.7568, cosine 鈮� -0.6536, tangent 鈮� 1.1584. For 70掳: sine 鈮� 0.9397, cosine 鈮� 0.3420, tangent 鈮� 2.7475. For 283掳: sine 鈮� -0.8829, cosine 鈮� 0.4695, tangent 鈮� -1.8807.

Step by step solution

Calculating Sine and Cosine for 0.15

To find the sine and cosine of 0.15, we use a calculator set in radian mode since the value is likely in radians. Use the calculator to find:$\sin(0.15) \approx 0.1494$$\cos(0.15) \approx 0.9888$

Calculating Tangent for 0.15

To calculate the tangent of 0.15, use the calculator again:$\tan(0.15) = \frac{\sin(0.15)}{\cos(0.15)} \approx \frac{0.1494}{0.9888} \approx 0.1511$

Calculating Sine and Cosine for 4

Check to ensure the calculator is in radian mode for computation:$\sin(4) \approx -0.7568$$\cos(4) \approx -0.6536$

Calculating Tangent for 4

Use the calculated values of sine and cosine:$\tan(4) = \frac{\sin(4)}{\cos(4)} \approx \frac{-0.7568}{-0.6536} \approx 1.1584$

Calculating Sine and Cosine for 70掳

Switch the calculator to degree mode. For 70掳:$\sin(70^{\circ}) \approx 0.9397$$\cos(70^{\circ}) \approx 0.3420$

Calculating Tangent for 70掳

Find tangent for 70掳 using sine and cosine:$\tan(70^{\circ}) = \frac{\sin(70^{\circ})}{\cos(70^{\circ})} \approx \frac{0.9397}{0.3420} \approx 2.7475$

Calculating Sine and Cosine for 283掳

Ensure the calculator is in degree mode:$\sin(283^{\circ}) \approx -0.8829$$\cos(283^{\circ}) \approx 0.4695$

Calculating Tangent for 283掳

Use the sine and cosine to find tangent:$\tan(283^{\circ}) = \frac{\sin(283^{\circ})}{\cos(283^{\circ})} \approx \frac{-0.8829}{0.4695} \approx -1.8807$

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Key Concepts

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

Sine Function

The sine function is one of the fundamental trigonometric functions, often denoted as $ \sin $. It relates to the y-coordinate of a point on the unit circle, which is a circle with a radius of one. The formula for sine is straightforward: for an angle $ \theta $ measured in radians or degrees from the positive x-axis, $ \sin(\theta) $ can be found as the length of the vertical leg (opposite side) in a right triangle. When using your calculator to find the sine of an angle, ensure that it's set to the correct mode (degrees or radians), depending on the units of the angle given. For example:

For an angle of 0.15 radians, adjusting the calculator to radian mode reveals that $ \sin(0.15) \approx 0.1494 $.
For 70 degrees, switch to degree mode to compute $ \sin(70^\circ) \approx 0.9397 $.

Rounding may slightly alter these values, so be mindful to use consistent precision.

Cosine Function

The cosine function, represented as $ \cos $, is another principal trigonometric function linked to the x-coordinate on the unit circle. The cosine of an angle$ \theta $ defines the length of the horizontal leg (adjacent side) of a right triangle. Similar to sine, it's essential to choose the correct mode on the calculator based on radians or degrees. Here's how it's generally calculated:

For 0.15 radians (often inferred as radians due to decimal format), $ \cos(0.15) \approx 0.9888 $.
Switching to degree mode for 283 degrees, the approximation becomes $ \cos(283^\circ) \approx 0.4695 $.

Cosine, like sine, can be negative or positive based on the quadrant of the angle, reflecting its symmetrical properties about the unit circle's center.

Tangent Function

The tangent function, denoted $ \tan $, is the ratio of the sine and cosine functions: $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $. It encapsulates how steep a line is when drawn from the origin to a point on the circle's perimeter. Tangent uses sine and cosine values to calculate:

At 4 radians, $ \tan(4) = \frac{\sin(4)}{\cos(4)} \approx 1.1584 $, determined in radian mode.
At 70 degrees, the angle in degree mode results in $ \tan(70^\circ) \approx 2.7475 $.

Because tangent is a ratio, it features distinct behavior, depending on the quadrants: positive in the first and third, yet negative in the second and fourth. These variations capture the angle's direction and nature on its applicable trigonometric circle.

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91影视

Use a calculator to find sine, cosine, and tangent of the following values: a. 0.15 b. 4 c. \(70^{\circ}\) d. \(283^{\circ}\)

Short Answer

Step by step solution

Calculating Sine and Cosine for 0.15

Calculating Tangent for 0.15

Calculating Sine and Cosine for 4

Calculating Tangent for 4

Calculating Sine and Cosine for 70掳

Calculating Tangent for 70掳

Calculating Sine and Cosine for 283掳

Calculating Tangent for 283掳

Key Concepts

Sine Function

Cosine Function

Tangent Function

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