# Precalculus : Trigonometric Functions

## Example Questions

### Example Question #4 : Find The Value Of The Sine Or Cosine Functions Of An Angle Given A Point On Its Terminal Side

If you are able to solve for the sine and cosine of an angle given a point on its terminal side, you have enough information to also solve for its tangent.

Which of the following best describes the validity of the above statement?

The statement is true in all cases.

The statement is false in all cases.

The statement is true in some cases, but not all.

The statement is true in all cases.

Explanation:

Recall the formula for tangent is .  If we have enough information to solve for sine and cosine, then we have the values of the opposite and adjacent of the angle.

### Example Question #5 : Find The Value Of The Sine Or Cosine Functions Of An Angle Given A Point On Its Terminal Side

The method of solving for trigonometric functions of an angle given a point on its terminal side only works for acute angles.

Which of the following statements best describes the validity of the statement above?

The statement is true

The statement is false

The statement is false

Explanation:

Consider the figure below.

We can form a triangle by dropping a line down from the point (-2,3) perpendicular to the

x axis.  Now we have right triangle that has a leg that is 3 units high and a base that is 2

units long.  Now we can use the Pythagorean Theorem to solve for the hypotenuse.

We can solve for sine and cosine now.  Use the fact that .  The opposite side of the angle in this case is the leg that is 3 units high.

And we will use the fact that .  The adjacent side to the angle is the base that is 2 units long.  We do have to note that we are using and not .

So even though our angle was obtuse, we can still use the same method.

### Example Question #6 : Find The Value Of The Sine Or Cosine Functions Of An Angle Given A Point On Its Terminal Side

Given that the cosine of an angle is , what is the height of the triangle formed by this?  (i.e. the terminal point for this angle is (1,y), solve for y).

Explanation:

We begin by thinking about what cosine means when working with right triangles.  Recall .  So if

Then the adjacent side, or the base of the triangle, is 1 and the hypotenuse is 2.  Now we can use the Pythagorean Theorem to solve for the missing side.

### Example Question #7 : Find The Value Of The Sine Or Cosine Functions Of An Angle Given A Point On Its Terminal Side

Find the value of the cosine of an angle given that the point on the terminal side of the angle is (3,4).

Explanation:

Using the point (3,4), we can see that this forms a right triangle that has a base that is 3 units in length and an adjoining leg that is 4 units high.  We are able to find the hypotenuse of this triangle using the Pythagorean Theorem.  This will give us the distance of the point (3,4) to the origin.

And so the hypotenuse of this triangle (the distance from our point we are working with to the origin), is 5 units long.  Recall that when using cosine for right triangles, cosine represents the following

So if we are considering the angle formed by the x-axis and our hypotenuse, the adjacent side would be the base of our triangle; 3 units.

### Example Question #8 : Find The Value Of The Sine Or Cosine Functions Of An Angle Given A Point On Its Terminal Side

Find the sine and cosine of the following angle.

Explanation:

We see that the point on the terminal side is (5,6).  So we know that with this point a right triangle is formed with a base that is 5 units long, and a leg that is 6 units high.  We can use the Pythagorean Theorem to solve for the hypotenuse that is formed by this triangle and this will tell us the distance of the point from the origin.

Let’s solve for sine first.  When working with right triangles recall that and we are considering the angle formed by the x-axis and the hypotenuse.  So the opposite side is the leg that is 6 units high.

We can solve for cosine if we recall that .  Our adjacent side would be the base that is 5 units long.

### Example Question #1 : Angular And Linear Velocity

If a ball is travelling in a circle of diameter  with velocity , find the angular velocity of the ball.

Explanation:

Using the equation,

where

=angular velocity, =linear velocity, and =radius of the circle.

In this case the radius is 5 (half of the diameter) and linear velocity is 20 m/s.

.

### Example Question #2 : Angular And Linear Velocity

Suppose a car tire rotates  times a second. The tire has a diameter of  inches. Find the angular velocity in radians per second.

Explanation:

Write the formula for angular velocity.

The frequency of the tire is 8 revolutions per second. The radius is not used.

Substitute the frequency and solve.

### Example Question #3 : Angular And Linear Velocity

What is the angular velocity of a spinning top if it travels  radians in a third of a second?

Explanation:

Write the formula for average velocity.

The units of omega is radians per second.

Substitute the givens and solve for omega.

### Example Question #4 : Angular And Linear Velocity

A  diamter tire on a car makes  revolutions per second. Find the angular speed of the car.

Explanation:

Recall that  .

Since the tire revolves 9.3 times/second it would seem that the tire would rotate

or .

We use  to indicate that the tire is rolling 360 degrees or  radians each revolution (as it should).

Thus,

Note that radians is JUST a different way of writing degrees. The higher numbers in the answers above are all measures around the actual linear speed of the tire, not the angular speed.

### Example Question #1 : Solve Angular Velocity Problems

A car wheel of radius 20 inches rotates at 8 revolutions per second on the highway. What is the angular speed of the tire?