# The Unit Circle: Sine and Cosine Functions

## Right Triangle Trigonometry


Mt. Everest, which straddles the border between China and Nepal, is the tallest mountain in the world. Measuring its height is no easy task. In fact, the actual measurement has been a source of controversy for hundreds of years. The measurement process involves the use of triangles and a branch of mathematics known as trigonometry. In this section, we will define a new group of functions known as trigonometric functions, and find out how they can be used to measure heights, such as those of the tallest mountains.


### Using Right Triangles to Evaluate Trigonometric Functions

 shows a right triangle with a vertical side of length 
 and a horizontal side has length 
 Notice that the triangle is inscribed in a circle of radius 1. Such a circle, with a center at the origin and a radius of 1, is known as a unit circle.
We can define the trigonometric functions in terms an angle t and the lengths of the sides of the triangle. The adjacent side is the side closest to the angle, x. (Adjacent means “next to.”) The opposite side is the side across from the angle, y. The hypotenuse is the side of the triangle opposite the right angle, 1. These sides are labeled in .
Given a right triangle with an acute angle of 
 the first three trigonometric functions are listed.
A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “ine is pposite over ypotenuse, osine is djacent over ypotenuse, angent is pposite over djacent.”
For the triangle shown in , we have the following.

### Reciprocal Functions

In addition to sine, cosine, and tangent, there are three more functions. These too are defined in terms of the sides of the triangle.
Take another look at these definitions. These functions are the reciprocals of the first three functions.
When working with right triangles, keep in mind that 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.
Many problems ask for all six trigonometric functions for a given angle in a triangle. A possible strategy to use is to find the sine, cosine, and tangent of the angles first. Then, find the other trigonometric functions easily using the reciprocals.

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

It is helpful to evaluate the trigonometric functions as they relate to the special angles—multiples of 
 and 
 Remember, however, 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, we 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 given 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 facilitates 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
