Problem 12 - Algebra 2

Question

Use a calculator and inverse functions to find the radian measures of the angles. angles whose cosine is \((-0.89)\)

Step-by-Step Solution

Answer

By taking the inverse cosine (arccos) of -0.89 on a calculator set to radian mode, the radian measure of the angle is obtained.

Step 1: Remember the inverse cosine

The first step is to understand the relationship between trigonometric ratios and their inverse. The arccosine, or inverse cosine function, is used to get the angle when we know the cosine of the angle. In this case, the cosine is given as -0.89.

Step 2: Use a calculator

This step involves using a scientific calculator to calculate the arccosine or inverse cosine of -0.89. Before doing this, ensure that the calculator is set to radian mode as the result is required in radian measure.

Step 3: Finding the radian

Input -0.89 into the calculator and apply the inverse cosine. The output will be the angle in radians.

Step 4: Write down the answer

The output value obtained is the required measure of the angle in radians. Make a note of this value as it is the solution.

Key Concepts

Radian Measure

When measuring angles, it's common to use degrees, but `radians` provide a different perspective. Radians are based on the radius of a circle. This unit is vital in trigonometry, calculus, and other mathematical fields. Understanding radian measure simplifies many equations and formulas because it relates directly to the radius of a circle:

• A full circle is equal to \(2\pi\) radians, equivalent to 360 degrees.
• Half a circle, or a straight angle, is \(\pi\) radians or 180 degrees.
• To convert degrees to radians, multiply the degree measure by \(\frac{\pi}{180}\).

Using radians in trigonometry allows for smoother calculations and integration with calculus. Always ensure your calculator is set to radian mode when asked for angles in radians.

Cosine Function

The `cosine function` is a fundamental trigonometric function representing the adjacent side over the hypotenuse in a right-angled triangle:

• The cosine function helps determine the angle when one side of a right triangle and the hypotenuse are known.
• Its value ranges from -1 to 1.
• Values like \(\cos(\theta)=0\) and \(\cos(\theta)=1\) have specific angle solutions: \(90^\circ\) and \(0^\circ\) respectively.

In the problem, the `inverse cosine` or `arccos` is utilized to find the angle whose cosine is -0.89. This involves knowing the cosine ratio, then finding the corresponding angle using a scientific calculator. It is crucial to understand that multiple angles can share the same cosine value, particularly in different quadrants.

Trigonometric Ratios

Trigonometric ratios relate the angles and sides of a right triangle. These ratios include sine, cosine, and tangent, each defined uniquely:

• `Sine` (sin) is the opposite side over the hypotenuse.
• `Cosine` (cos) is the adjacent side over the hypotenuse.
• `Tangent` (tan) is the opposite side over the adjacent side.

Knowing these ratios helps solve various mathematical problems, including finding angles when given certain side lengths. The ability to switch between these ratios and their inverse counterparts, such as arcsin, arccos, and arctan, allows for solving for angles directly from ratios. In this exercise, the focus is on using the inverse cosine, a trigonometric ratio, to find the angle in radians.

Scientific Calculator

A `scientific calculator` is indispensable for trigonometric calculations, especially when finding angles in radians. These calculators have built-in functions for trigonometric ratios and their inverses:

• Make sure your calculator is in `radian mode` when working with radian measures to avoid incorrect results.
• Understand how to access the `inverse trigonometric functions`, often through a "shift" or "second" key on the calculator.
• For this problem, knowing where the `arccos` or `cos⁻¹` function is will directly lead to finding the angle whose cosine value is known.

Using a scientific calculator enhances accuracy and efficiency in these computations, ensuring you reach a solution quickly and correctly.