# Trigonometry in the Right Triangle and with the Unit Circle

## Trigonometric Relations in the Right Triangle

A right angle is an angle that measures {eq}90^{\circ} {/eq} (or {eq}\frac{\pi}{2} {/eq} if we want to give the measure in radians). Throughout this lesson, we are not going to stick to one unit to give the measure of an angle.

Now, a right triangle is a triangle with one of the interior angles measuring {eq}90^{\circ}, {/eq} like the triangle {eq}ABC {/eq} with sides {eq}a, b {/eq} and {eq}c {/eq} in the following figure.

The biggest side, opposed to the right angle, is called the hypothenuse, and the other two sides are the legs. In a right triangle, sine of an angle is defined as the ratio between the opposite side and the hypothenuse, cosine is the ratio between the adjacent side and the hypothenuse, and tangent is the ratio between the sine and cosine of the angle, or, equivalently, the ratio between the opposite side and the adjacent one. Defining the trigonometric ratios in the triangle {eq}ABC {/eq} for {eq}\beta {/eq} gives

$$\sin \beta = \frac{a}{b}, \cos \beta = \frac{c}{b} ~ \textrm{and} ~ \tan \beta = \frac{a}{c} $$

## Unit Circle Triangles

The idea of sine and cosine can be applied for every real number, and not only for the angles of a right triangle. The notion of a tangent, however, cannot be extended to some numbers because it is the ratio between the sine and the cosine. Because it is a fraction, the denominator cannot be zero. When we expand the trigonometric concepts, it is necessary to use a unit circle, which is simply a circle of radius {eq}1 {/eq} centered at the origin {eq}O(0, 0), {/eq} as shown in the image.

The coordinate axes divide the Cartesian plane into four regions, called quadrants, and defined in the following list, which contains a fancy symbolic notation that will be explained afterward:

- First quadrant: {eq}Q_1 = \{(x, y) \in \mathbb{R}^2 | x > 0, y > 0\} {/eq}

- Second quadrant: {eq}Q_2 = \{(x, y) \in \mathbb{R}^2 | x < 0, y > 0\} {/eq}

- Third quadrant: {eq}Q_3 = \{(x, y) \in \mathbb{R}^2 | x < 0, y < 0\} {/eq}

- Fourth quadrant: {eq}Q_4 = \{(x, y) \in \mathbb{R}^2 | x > 0, y < 0\} {/eq}

The first quadrant is the set of points in the Cartesian plane where both the abscissa and the ordinate are positive. The explanation of the definition of the other quadrants is analogous and, for this reason, will be omitted. In the next image, we show the four quadrants and the coordinate axes.

Let's see a couple of **unit circle triangles** in a figure because they will help us when defining the trigonometric functions.

Observe the triangles {eq}OPR {/eq} and {eq}OQS. {/eq} We are going to refer to right triangles formed by the origin, a point on the circumference, and points on the coordinate axes as unit circle triangles. They will be helpful when defining trigonometric functions. Rays {eq}\overrightarrow{OP} {/eq} and {eq}\overrightarrow{OR} {/eq} form two angles, as well as {eq}\overrightarrow{OR} {/eq} and {eq}\overrightarrow{OQ}. {/eq} In trigonometry, we adopt the convention that a counterclockwise rotation beginning on the positive x-axis gives a positive angle, whereas a clockwise one, a negative angle.

### Trig Functions Using the Unit Circle

Now we are ready to see the definitions of the trig functions using the **trig unit circle**. Consider the last example where we have a right triangle {eq}OPR, {/eq} with {eq}P(x, y) {/eq} being a point on the circumference and {eq}\alpha {/eq} being the angle depicted below.

Using the definition of sine, cosine, and tangent that we gave at the beginning for angles in the right triangle we get: {eq}\sin \alpha = \frac{y}{1}, \cos \alpha = \frac{x}{1} ~ \textrm{and} ~ \tan \alpha = \frac{y}{x} {/eq} In general, a point on the unit circle will determine an angle with the positive x-axis and the abscissa of such point will be the angle's cosine, the ordinate will be the angle's sine, and the ratio of the two will be the angle's tangent (provided that the cosine is not zero). In particular, for the angles {eq}0 {/eq} and {eq}\frac{\pi}{2} {/eq} the points on the unit circle will be {eq}(1, 0) {/eq} and {eq}(0, 1), {/eq} respectively. Thus, {eq}\sin 0 = 0, \sin \frac{\pi}{2} = 1, \cos 0 = 1 ~ \textrm{and} ~ \cos \frac{\pi}{2} = 0. {/eq}

#### Sine of an Angle Using the Unit Circle

Because the sine of a number is the ordinate of a point on the unit circle and, if we walk counterclockwise on it from the point {eq}(1, 0), {/eq} the conclusion is that the sine is increasing on the first and fourth quadrants, and decreasing on the second and third ones. On the first and second quadrants, sine will be positive; it will be negative on the third and fourth quadrants. Also, the maximum y-value of a point on the unit circle is {eq}1 {/eq}, and the minimum, {eq}-1. {/eq} So, the range of the sine function is the interval {eq}[-1, 1]. {/eq}

#### Cosine of an Angle Using the Unit Circle

The results are analogous to the cosine function. The cosine of a number is associated with the abscissa of a point on the unit circle. If we walk counterclockwise on the unit circle from the point {eq}(1, 0), {/eq} the abscissa will decrease on the first quadrant and second as well; and it will increase on the third and fourth quadrants. The cosine function will be positive on the first and fourth quadrants, and negative on the second and third. Furthermore, the range of the cosine function is the interval {eq}[-1, 1]. {/eq}

#### Unit Circle: Sine, Cosine, and Tangent

As defined at the beginning of the lesson, the tangent of an angle is the ratio between the sine and the cosine. Because of this, when the cosine of an angle is zero, the tangent will be undefined. Based on the above information, the conclusion is that the tangent will be positive on the first and third quadrants, and negative on the other two. Because the value of cosine gets close to zero with positive and negative numbers, the tangent goes to positive and negative infinity. Therefore, the interval of the tangent function is {eq}(-\infty, \infty). {/eq}

## Important Angles

The angles of {eq}30^{\circ}, 45^{\circ} {/eq} and {eq}60^{\circ} {/eq} are the three important ones, whose values are displayed in the following table. To memorize the sine and cosine of those notable angles, keep in mind that all are fractions with the denominator {eq}2. {/eq} Then, the numerators of sine are {eq}1, 2, 3, {/eq} and {eq}3, 2, 1 {/eq} the numerators of cosine.

x | 30 | 45 | 60 |
---|---|---|---|

sin (x) | 1/2 | sqrt(2)/2 | sqrt(3)/2 |

cos (x) | sqrt(3)/2 | sqrt(2)/2 | 1/2 |

tan (x) | sqrt(3)/3 | 1 | sqrt(3) |

### Example 1: How to Find Sine of a Right Triangle?

Determine the angle {eq}\alpha {/eq} based on the following image.

We have {eq}\sin \alpha = \frac{3}{6} = \frac{1}{2}. {/eq} Checking the table from the previous section, we see that {eq}\alpha = 30^{\circ}. {/eq}

### Example 2: What is Cos(330{eq}^{\circ} {/eq})?

To find {eq}\cos 330^{\circ}, {/eq} let's start with the image of the angle in the unit circle.

There is a congruent triangle to the one in the image in the first quadrant, associated with the angle of {eq}30^{\circ}. {/eq} Because the cosine is positive in the first and fourth quadrants, {eq}\cos 330^{\circ} = \frac{\sqrt{3}}{2}. {/eq}

## Lesson Summary

In this lesson we saw the sine (opposite side over hypothenuse), cosine (adjacent side over hypotenuse), and tangent (sine over cosine) of angles, beginning with trigonometric relations in the right triangle and expanding those notions to the **trig unit circle**, a circle centered at the origin with a radius of one. We showed the trigonometric ratios of important angles and of {eq}0^{\circ} {/eq} and {eq}90^{\circ} {/eq} too: {eq}\sin 0 = 0, \cos 0 = 1, \sin 90^{\circ} = 1, \cos 90^{\circ} = 0. {/eq} The lesson ended with a couple of examples where we saw how to find the sine using a right triangle and the cosine using **unit circle triangles**.

To unlock this lesson you must be a Study.com Member.

Create your account

#### What is the unit circle in trigonometry?

In trigonometry, the unit circle is a circle of radius 1 centered at the origin, that is, the point (0, 0)

#### What is cos on the unit circle?

Let O(0, 0) and P(x, y) be a point on the unit circle. The cosine of the angle between the positive x-axis and OP is x, the abscissa of P.

### Register to view this lesson

### Unlock Your Education

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.

Become a MemberAlready a member? Log In

BackAlready registered? Log in here for access