Problem 101 - Precalculus

Question

Calculate sce \(70^{\circ}\) in the following two ways: a. Find \(\cos 70^{\circ}\) to three decimal places and then divide 1 by that number. Write that number to five decimal places. b. With a calculator in degree mode, enter \(70, \cos , 1 / \mathrm{x},\) and round the result to five decimal places.

Step-by-Step Solution

Answer

\(\sec 70^{\circ} \approx 2.92398\) Calculated both by inversion of \(\cos 70^{\circ}\) and direct calculator method.

Step 1: Calculate Cosine

Find the cosine of 70° to three decimal places using a calculator. \[ \cos 70^{\circ} \approx 0.342 \]

Step 2: Invert the Cosine

Calculate the secant by dividing 1 by the value of the cosine obtained in Step 1.\[ \sec 70^{\circ} = \frac{1}{\cos 70^{\circ}} = \frac{1}{0.342} \approx 2.923976 \]

Step 3: Calculator Direct Method

Set the calculator to degree mode and enter the following sequence: 70, cos, 1/x. The calculator will give the same output: approximately 2.92398 when rounded to five decimal places.

Key Concepts

Cosine Function

The cosine function is a fundamental part of trigonometry, which deals with angles and distances. It's a ratio that helps to determine the horizontal component of an angle in a right triangle, working with the adjacent and hypotenuse sides.
When you see something like \( \cos 70^{\circ} \), it refers to the cosine of a 70-degree angle. This specific function, like others in trigonometry, is derived from a unit circle – a circle with a radius of 1. In this circle, each angle corresponds to a point and its coordinates, where cosine represents the x-coordinate of the point on the circle.
Let's make it simple:

• The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
• For example, if \( \cos 70^{\circ} \approx 0.342 \), it means that the adjacent side is about 34.2% the length of the hypotenuse in a right triangle with that angle.
• Cosine is a periodic function, repeating its values every 360 degrees, and it can take values only between -1 and 1.

Cosine occasionally confuses students because it's abstract, but keep picturing the unit circle and right triangles. It becomes clearer with practice.

Trigonometric Identities

Trigonometric identities are like mathematical "rules" that relate different trigonometric functions to one another. These identities are essential tools for simplifying expressions and solving equations involving trigonometric functions.
Among the key identities, you'll often use:

• Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Reciprocal Identities: These express one function as the inverse of another. For example, the secant function, denoted as \( \sec(\theta) \), is the reciprocal of the cosine: \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• Angle Sum and Difference Identities: These help calculate the cosine or sine of combined angles, such as \( \cos(\alpha \pm \beta) \).

For our task, we used the reciprocal identity. When finding \( \sec 70^{\circ} \), we calculated it by taking the reciprocal of \( \cos 70^{\circ} \) – the essence of such identities is to turn complex-looking problems into simple arithmetic operations.

Calculator Usage

Using a calculator properly is an essential skill, especially in trigonometry where precision is required. Calculators help perform complex calculations quickly and accurately, but only when set up correctly.
Here's how you can use them for trigonometric calculations:

• Mode Setting: Always ensure your calculator is in the correct mode. For trigonometry, it should be in degree mode if you're working with degrees (like our problem with \(70^{\circ}\)).
• Calculate Cosine: Enter the angle and press the cosine button to get the value of \( \cos(\theta) \).
• Use the Invert Function: Instead of manually dividing 1 by the cosine result, use the \( 1/x \) button after calculating \( \cos(70^{\circ}) \). This function cleverly computes the reciprocal, making secant calculations fast and accurate.

Remember, calculators do what you tell them, not what you mean. Double-check your settings and sequences to avoid unnecessary errors.