Problem 5

Question

Calculate (be sure that your calculator is in radian or degree mode as needed). (a) $\frac{56.4 \tan 34.2^{\circ}}{\sin 34.1^{\circ}}$ (b) $\frac{5.34 \tan 21.3^{\circ}}{\sin 3.1^{\circ}+\cot 23.5^{\circ}}$ (c) $\tan 0.452$ (d) $\sin (-0.361)$

Step-by-Step Solution

Verified

Answer

(a) Calculate using degrees; (b) calculate using degrees; (c) and (d) calculate using radians.

1Step 1: Understand the Problem

We need to calculate the given trigonometric expressions. Ensure your calculator is in the correct mode (degree for parts (a) and (b), radian for parts (c) and (d)).

2Step 2: Solve Part (a)

Compute the value of $ \tan 34.2^{\circ} $ and $ \sin 34.1^{\circ} $ using your calculator in degree mode. Then calculate $ \frac{56.4 \times \tan 34.2^{\circ}}{\sin 34.1^{\circ}} $.

3Step 3: Solve Part (b)

Compute $ \tan 21.3^{\circ} $, $ \sin 3.1^{\circ} $, and $ \cot 23.5^{\circ} = \frac{1}{\tan 23.5^{\circ}} $ in degree mode. Substitute these into $ \frac{5.34 \times \tan 21.3^{\circ}}{\sin 3.1^{\circ} + \cot 23.5^{\circ}} $ and simplify.

4Step 4: Solve Part (c)

Set your calculator to radian mode and find $ \tan 0.452 $.

5Step 5: Solve Part (d)

Make sure your calculator is in radian mode and find $ \sin(-0.361) $.

Key Concepts

Degree ModeRadian ModeTrigonometric FunctionsCalculator Usage

Degree Mode

Understanding degree mode is crucial when working with trigonometric functions that involve angle measurements in degrees. Degree mode is a setting on your calculator that allows you to input angles as you would see them on a protractor or compass, ranging from 0° to 360°.
When using degree mode:

Make sure the calculator displays degree symbols, usually marked as "DEG" in the settings.
Your input for angles will be interpreted in degrees rather than radians.
Common applications are solving problems involving compass directions, slopes, and navigation.

Before solving a problem, double-check that your calculator is set to degree mode if the problem provides angles in degrees. Incorrect mode settings can lead to significant errors in computations.

Radian Mode

When an exercise involves radian mode, it's about working with angles measured in radians rather than degrees. A radian is another way to measure angles, commonly used in calculus and higher-level mathematics due to its natural connection to the radius of a circle.
In radian mode:

The calculator will interpret angle inputs as radians, which are unitless numbers.
One complete revolution around a circle is 2π radians, equivalent to 360 degrees.
Understanding radians is essential for applications in physics and engineering.

Ensure your calculator is in radian mode by checking for an indicator, often marked "RAD." Many problems, especially in calculus, require calculations in radians, so being comfortable with switching modes is vital.

Trigonometric Functions

Trigonometric functions, fundamental in mathematics, help relate angles to side lengths in right triangles and model periodic phenomena.
Key trigonometric functions include:

Sine ($\sin$): It gives the ratio of the opposite side to the hypotenuse in a right triangle.
Cosine ($\cos$): This ratio is of the adjacent side to the hypotenuse.
Tangent ($\tan$): It equals the ratio of the opposite side to the adjacent side.
Cotangent ($\cot$): This is the reciprocal of tangent, $\cot θ = \frac{1}{\tan θ}$.

These functions appear in various applications from calculating angles and distances in geometry to analyzing wave patterns in physics.

Calculator Usage

Using a calculator effectively is essential for solving trigonometric problems. Calculators can quickly compute values of trigonometric functions once set in the correct mode.
Steps for proper calculator usage:

Always check whether the calculator is in "DEG" or "RAD" mode, depending on whether you are working with degrees or radians.
Input the trigonometric expressions exactly as shown, ensuring correct use of parentheses.
Practice switching between modes, as many problems require using both degrees and radians at different stages.
Use the calculator's function keys like, $\sin^{-1}, \cos^{-1}, \tan^{-1}$, for inverse operations when needed.

Familiarity with your calculator model's layout and settings is important. Regular practice will enhance accuracy and confidence in calculations.

Problem 5

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