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# 2.1: Trigonometric Ratios

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Trigonometry, Chapter 1, Lesson 3.

In this activity, you will explore:

• Sine, cosine and tangents of angles
• Side length ratios of a right triangle
• Finding the missing side of a triangle when an angle and side are given.

Before beginning this activity, make sure your calculator is in DEGREE mode. Press the MODE button, arrow down two lines, arrow to the right to select DEGREE, and press ENTER.

## Problem 1 – Trigonometric Ratios

Start the CabriJr app by pressing the APPS button and choosing it from the menu.

Select Open from the F1: File menu, and choose the file TRIG.

In right triangles, there is a relationship between the ratios of the side lengths and the trigonometric functions.

Find the ratios below using the values listed for the graph. Then find the values for the trig functions below using the value given for A\begin{align*}A\end{align*} on the graph.

BCAC=,ACAB=,BCAB=\begin{align*}\frac{BC}{AC}=\frac{\;\;\;\;\;\;\;\;\;}{\;\;\;\;\;\;\;\;\;}, \frac{AC}{AB}=\frac{\;\;\;\;\;\;\;\;\;}{\;\;\;\;\;\;\;\;\;}, \frac{BC}{AB}=\frac{\;\;\;\;\;\;\;\;\;}{\;\;\;\;\;\;\;\;\;}\end{align*}

Sin A=,Cos A=,Tan A=\begin{align*}\text{Sin} \ A = \underline{\;\;\;\;\;\;\;\;\;}, \text{Cos} \ A = \underline{\;\;\;\;\;\;\;\;\;}, \text{Tan} \ A = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Repeat this for two more different triangles by moving point B to a different location (to grab a point, press the ALPHA button and use the arrow keys to move it.)

Triangle #2

BCAC=,ACAB=,BCAB=\begin{align*}\frac{BC}{AC}=\frac{\;\;\;\;\;\;\;\;\;}{\;\;\;\;\;\;\;\;\;}, \frac{AC}{AB}=\frac{\;\;\;\;\;\;\;\;\;}{\;\;\;\;\;\;\;\;\;}, \frac{BC}{AB}=\frac{\;\;\;\;\;\;\;\;\;}{\;\;\;\;\;\;\;\;\;}\end{align*}

Sin A=,Cos A=,Tan A=\begin{align*}\text{Sin} \ A = \underline{\;\;\;\;\;\;\;\;\;}, \text{Cos} \ A = \underline{\;\;\;\;\;\;\;\;\;}, \text{Tan} \ A = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Triangle #3

BCAC=,ACAB=,BCAB=\begin{align*}\frac{BC}{AC}=\frac{\;\;\;\;\;\;\;\;\;}{\;\;\;\;\;\;\;\;\;}, \frac{AC}{AB}=\frac{\;\;\;\;\;\;\;\;\;}{\;\;\;\;\;\;\;\;\;}, \frac{BC}{AB}=\frac{\;\;\;\;\;\;\;\;\;}{\;\;\;\;\;\;\;\;\;}\end{align*}

Sin A=,Cos A=,Tan A=\begin{align*}\text{Sin} \ A = \underline{\;\;\;\;\;\;\;\;\;}, \text{Cos} \ A = \underline{\;\;\;\;\;\;\;\;\;}, \text{Tan} \ A = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Based upon your answers hypothesize which ratio goes with each trigonometric function.

Sin A=Cos A=Tan A=\begin{align*}\text{Sin} \ A = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \quad \text{Cos} \ A = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \quad \text{Tan} \ A = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

A good acronym to use to help remember these relationships is SOHCAHTOA

sin A=OppositeHypotenusecos A=AdjacentHypotenusetan A=OppositeAdjacent\begin{align*}&\sin \ A=\frac{\text{Opposite}}{\text{Hypotenuse}}\\ &\cos \ A=\frac{\text{Adjacent}}{\text{Hypotenuse}}\\ &\tan \ A=\frac{\text{Opposite}}{\text{Adjacent}}\end{align*}

## Problem 2 – Trigonometry, What Is It Good For?

One of the uses of trigonometry is finding missing side lengths of a triangle.

To find the length of side BC\begin{align*}BC\end{align*} in the triangle to the right, write the sine relationship.

Find BC\begin{align*}BC\end{align*}.

Now solve for BC\begin{align*}BC\end{align*} and calculate using your handheld.

To find the length of side AC\begin{align*}AC\end{align*} in the triangle to the right, write the cosine relationship.

Find AC\begin{align*}AC\end{align*}.

Now solve for AC\begin{align*}AC\end{align*} and calculate using your handheld.

To find the length of side AC\begin{align*}AC\end{align*} in the triangle to the right, write the tangent relationship.

Find AC\begin{align*}AC\end{align*}.

Now solve for AC\begin{align*}AC\end{align*} and calculate using your handheld.

## Student Exercises

Write the correct trigonometric function for each triangle below and solve for the missing side.

1. Find AC\begin{align*}AC\end{align*}.

2. Find BC\begin{align*}BC\end{align*}.

3. Find AC\begin{align*}AC\end{align*}.

4. Find AC\begin{align*}AC\end{align*}.

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