# Trigonometric Functions

## Right Triangle Trigonometry

We have previously defined the sine and cosine of an angle in terms of the coordinates of a point on the unit circle intersected by the terminal side of the angle:
In this section, we will see another way to define trigonometric functions using properties of right triangles.

### Using Right Triangles to Evaluate Trigonometric Functions

In earlier sections, we used a unit circle to define the trigonometric functions. In this section, we will extend those definitions so that we can apply them to right triangles. The value of the sine or cosine function of 
  is its value at 
  radians. First, we need to create our right triangle.  shows a point on a unit circle of radius 1. If we drop a vertical line segment from the point 
  to the x-axis, we have a right triangle whose vertical side has length 
  and whose horizontal side has length 
  We can use this right triangle to redefine sine, cosine, and the other trigonometric functions as ratios of the sides of a right triangle.
We know
Likewise, we know
These ratios still apply to the sides of a right triangle when no unit circle is involved and when the triangle is not in standard position and is not being graphed using 
  coordinates. To be able to use these ratios freely, we will give the sides more general names: Instead of 
  we will call the side between the given angle and the right angle the adjacent side to angle 
  (Adjacent means “next to.”) Instead of 
  we will call the side most distant from the given angle the opposite side from angle 
  And instead of 
  we will call the side of a right triangle opposite the right angle the hypotenuse. These sides are labeled in .

### Understanding Right Triangle Relationships

Given a right triangle with an acute angle of  
 

A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, Tangent is opposite over adjacent.”

### Relating Angles and Their Functions

When working with right triangles, the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in . The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.
We will be asked to find all six trigonometric functions for a given angle in a triangle. Our strategy is to find the sine, cosine, and tangent of the angles first. Then, we can find the other trigonometric functions easily because we know that the reciprocal of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent.

### Finding Trigonometric Functions of Special Angles Using Side Lengths

We have already discussed the trigonometric functions as they relate to the special angles on the unit circle. Now, we can use those relationships to evaluate triangles that contain those special angles. We do this because when we evaluate the special angles in trigonometric functions, they have relatively friendly values, values that contain either no or just one square root in the ratio. Therefore, these are the angles often used in math and science problems. We will use multiples of  
  
  and 
 
however, remember that when dealing with right triangles, we are limited to angles between 
 

Suppose we have a 
  triangle, which can also be described as a 
  triangle. The sides have lengths in the relation 
  The sides of a 
  triangle, which can also be described as a 
  triangle, have lengths in the relation 
  These relations are shown in .
We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.

### Using Equal Cofunction of Complements 

If we look more closely at the relationship between the sine and cosine of the special angles relative to the unit circle, we will notice a pattern. In a right triangle with angles of 
  and 
  we see that the sine of 
  namely 
  is also the cosine of 
  while the sine of 
  namely 
  is also the cosine of 
 

See 
This result should not be surprising because, as we see from , the side opposite the angle of 
  is also the side adjacent to 
  so 
  and 
  are exactly the same ratio of the same two sides, 
  and 
  Similarly, 
  and 
  are also the same ratio using the same two sides, 
  and 
 

The interrelationship between the sines and cosines of 
  and 
  also holds for the two acute angles in any right triangle, since in every case, the ratio of the same two sides would constitute the sine of one angle and the cosine of the other. Since the three angles of a triangle add to 
  and the right angle is 
  the remaining two angles must also add up to 
  That means that a right triangle can be formed with any two angles that add to 
  —in other words, any two complementary angles. So we may state a cofunction identity: If any two angles are complementary, the sine of one is the cosine of the other, and vice versa. This identity is illustrated in .
Using this identity, we can state without calculating, for instance, that the sine of 
  equals the cosine of 
  and that the sine of 
  equals the cosine of 
  We can also state that if, for a certain angle 
 
 
  then 
  as well.

### Using Trigonometric Functions

In previous examples, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.

### Using Right Triangle Trigonometry to Solve Applied Problems

Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height. Similarly, we can form a triangle from the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. See .

### Key Equations


### Key Concepts

1. We can define trigonometric functions as ratios of the side lengths of a right triangle. See .
2. The same side lengths can be used to evaluate the trigonometric functions of either acute angle in a right triangle. See .
3. We can evaluate the trigonometric functions of special angles, knowing the side lengths of the triangles in which they occur. See .
4. Any two complementary angles could be the two acute angles of a right triangle.
5. If two angles are complementary, the cofunction identities state that the sine of one equals the cosine of the other and vice versa. See .
6. We can use trigonometric functions of an angle to find unknown side lengths.
7. Select the trigonometric function representing the ratio of the unknown side to the known side. See .
8. Right-triangle trigonometry permits the measurement of inaccessible heights and distances.
9. The unknown height or distance can be found by creating a right triangle in which the unknown height or distance is one of the sides, and another side and angle are known. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, use cofunctions of complementary angles.
For the following exercises, find the lengths of the missing sides if side 
  is opposite angle 
  side 
  is opposite angle 
  and side 
  is the hypotenuse.

### Graphical

For the following exercises, use  to evaluate each trigonometric function of angle 
 

For the following exercises, use  to evaluate each trigonometric function of angle 
 

For the following exercises, solve for the unknown sides of the given triangle.

### Technology

For the following exercises, use a calculator to find the length of each side to four decimal places.

### Extensions


### Real-World Applications


### Review Exercises


### Angles

For the following exercises, convert the angle measures to degrees.
For the following exercises, convert the angle measures to radians.
For the following exercises, find the angle between 0° and 360° that is coterminal with the given angle.
For the following exercises, find the angle between 0 and 
  in radians that is coterminal with the given angle.
For the following exercises, draw the angle provided in standard position on the Cartesian plane.

### Unit Circle: Sine and Cosine Functions


### The Other Trigonometric Functions

For the following exercises, find the exact value of the given expression.
For the following exercises, use reference angles to evaluate the given expression.

### Right Triangle Trigonometry

For the following exercises, use side lengths to evaluate.
For the following exercises, use the given information to find the lengths of the other two sides of the right triangle.
For the following exercises, use  to evaluate each trigonometric function.
For the following exercises, solve for the unknown sides of the given triangle.

### Practice Test
