# Chapter 1: Right Triangles and an Introduction to Trigonometry

**At Grade**Created by: CK-12

- 1.1.
## The Pythagorean Theorem

- 1.2.
## Special Right Triangles

- 1.3.
## Basic Trigonometric Functions

- 1.4.
## Solving Right Triangles

- 1.5.
## Measuring Rotation

- 1.6.
## Applying Trig Functions to Angles of Rotation

- 1.7.
## Trigonometric Functions of Any Angle

- 1.8.
## Relating Trigonometric Functions

### Chapter Summary

In this chapter students learn about right triangles and special right triangles. Trigonometry is discovered through the special right triangles and the Pythagorean Theorem. Sine, cosine, tangent, secant, cosecant, and cotangent are all functions of angles and the result is the ratio of the sides of a right triangle. Only our special right triangles generate sine, cosine, tangent values that can be found without the use of a scientific calculator. When incorporating the trig ratios and the Pythagorean Theorem, we discover the first of many trig identities. This concept is explored further in Chapter 3.

## Vocabulary

- Adjacent
- A side adjacent to an angle is the side next to the angle. In a right triangle, it is the leg that is next to the angle.

- Angle of depression
- The angle between the horizontal line of sight, and the line of sight down to a given point.

- Angle of elevation
- The angle between the horizontal line of sight, and the line of sight up to a given point.

- Bearings
- The direction from one object to another, usually measured as an angle.

- Clinometer
- A device used to measure angles of elevation or depression.

- Coterminal angles
- Two angles in standard position are coterminal if they share the same terminal side.

- Distance Formula
- \begin{align*}d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\end{align*}

- Hypotenuse
- The hypotenuse is the longest side in a right triangle, opposite the right angle.

- Identity
- An identity is an equation that is always true, as long as the variables and expressions involved are defined.

- Included Angle
- The angle inbetween two sides of a polygon.

- Leg
- The legs of a right triangle are the two shorter sides.

- Nautical Mile
- A nautical mile is a unit of length that corresponds approximately to one minute of latitude along any meridian. A nautical mile is equal to 1.852 meters.

- Pythagorean Theorem
- \begin{align*}a^2 + b^2 = c^2\end{align*}

- Pythagorean Triple
- A set whole numbers for which the Pythagorean Theorem holds true.

- Quadrantal angle
- A quadrantal angle is an angle in standard position whose terminal side lies on an axis.

- Radius
- The radius of a circle is the distance from the center of the circle to the edge. The radius defines the circle.

- Reference angle
- The reference angle of an angle in standard position is the measure of the angle between the terminal side and the closest portion of the \begin{align*}x-\end{align*}axis.

- Standard position
- An angle in standard position has its initial side on the positive \begin{align*}x-\end{align*}axis, its vertex at the origin, and its terminal side anywhere in the plane. A positive angle means a counterclockwise rotation. A negative angle means a clockwise rotation.

- Theodolite
- A device used to measure angles of elevation or depression.

- Unit Circle
- The unit circle is the circle with radius 1 and center (0, 0). The equation of the unit circle is \begin{align*}x^2 + y^2 = 1\end{align*}

## Review Questions

- One way to prove the Pythagorean Theorem is through the picture below. Determine the area of the square two different ways and set each equal to each other.
- A flute is resting diagonally, \begin{align*}d\end{align*}, in the rectangular box (prism) below. Find the length of the flute.
- Solve the right triangle.
- Solve the right triangle.
- Find the
*exact*value of the area of the parallelogram below. - The modern building shown below is built with an outer wall (shown on the left) that is not at a 90-degree angle with the floor. The wall on the right is perpendicular to both the floor and ceiling. Find the length of \begin{align*}w\end{align*}.
- Given that \begin{align*}\cos(90^\circ-x) = \frac{2}{7}\end{align*}, find the \begin{align*}\sin x\end{align*}.
- If \begin{align*}\cos(-x) = \frac{3}{4}\end{align*} and \begin{align*}\tan x = \frac{\sqrt{7}}{3}\end{align*}, find \begin{align*}\sin(-x)\end{align*}.
- If \begin{align*}\sin y = \frac{1}{3}\end{align*}, what is \begin{align*}\cos y\end{align*}?
- \begin{align*}\sin \theta = \frac{1}{3}\end{align*} find the value(s) of \begin{align*}\cos \theta\end{align*}.
- \begin{align*}\cos \theta = -\frac{2}{5}\end{align*}, and \begin{align*}\theta\end{align*} is a second quadrant angle. Find the exact values of remaining trigonometric functions.
- (3, -4) is a point on the terminal side of \begin{align*}\theta\end{align*}. Find the exact values of the six trigonometric functions.
- Determine reference angle and two coterminal angles for \begin{align*}165^\circ\end{align*}. Plot the angle in standard position.
- It is very helpful to have the unit circle with all the special values on one circle. Fill out the unit circle below with all of the endpoints for each special value and quadrantal value.

## Texas Instruments Resources

*In the CK-12 Texas Instruments Trigonometry FlexBook® resource, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9699.*

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