Trig Values of Any Angle Objective: To define trig values for any angle; to construct the “Unit Circle.”

Angles Lets look at any point. We can use our knowledge of the six trig functions to make a general statement.

Angles Lets look at any point. We can use our knowledge of the six trig functions to make a general statement.

Example 1 Let (-3,4) be a point of the terminal side of  Find the sin, cos, and tan of 

Example 1 Let (-3,4) be a point of the terminal side of  Find the sin, cos, and tan of  This means that x = -3, y = 4, and by the Pythagorean Theorem, r = 5.

You Try Find all six trig functions for the following angle. You need to find the value of x first.

You Try Find all six trig functions for the following angle. You need to find the value of x first.

Sign Charts The sign of the sin, cos, and tan functions depends on which quadrant the angle is in. Since the radius is always positive, the sign of the x and y coordinates will determine the sign of the trig function.

Sign Charts The sin function is positive where y is positive and negative where y is negative. The cos function is positive where the x is positive and negative where x is negative. The tangent function is positive where x and y are the same sign and negative where they have opposite signs.

Sign Charts This is the sign chart for the sin, cos, and tan functions. Memorize these, you will need to know them at all times.

Quadrant Angles The benefit of using a unit circle is that the sin value is just the y coordinate and the cos is the x coordinate. This leads us to the following:

The Unit Circle We will now look at all angles on what we call the Unit Circle. This is a circle with a radius of 1. This will make all of our work much easier. Since the radius is 1, the sin just becomes the y coordinate and the cos is the x coordinate.

Reference Angles We are going to use reference angles to find values. A reference angle is the acute angle  formed by the terminal side of  and the horizontal axis. In other words, it is the distance to the x axis.

Reference Angles We are going to use reference angles to find values. A reference angle is the acute angle  formed by the terminal side of  and the horizontal axis. In other words, it is the distance to the x axis.

Example 4 Find the reference angle for . a)  = b)  = c)  =

Example 4 Find the reference angle for . a)  = b)  = c)  = a)Distance from the x-axis is 30. b)Distance from the x-axis is 60. c)Distance from the x-axis is 45.

Example 4 Find the reference angle for . a)  = a)When the angle is greater than 270 0, we are looking for the distance to 360 0, not The reference angle for this is 60 0.

Example 4 Find the reference angle for . We can also do this in radians. We are looking for the distance from  a)  = 3  /4 b)  = 7  /6 c)  = -5  /3

Example 4 Find the reference angle for . We can also do this in radians. We are looking for the distance from  a)  = 3  /4 b)  = 7  /6 c)  = -5  /3 a)The distance from  is  /4. b)The distance from  is  /6. c)The distance from  is  /3.

Using Reference Angles Using the three angles in the first quadrant, we can find the exact value of several other angles. Lets look at the reference angle of 30 0, or  /6.

Using Reference Angles Using the three angles in the first quadrant, we can find the exact value of several other angles. Lets look at the reference angle of 30 0, or  /6.

Using Reference Angles Using the three angles in the first quadrant, we can find the exact value of several other angles. Lets look at the reference angle of 60 0, or  /3.

Using Reference Angles Using the three angles in the first quadrant, we can find the exact value of several other angles. Lets look at the reference angle of 60 0, or  /3.

Using Reference Angles Using the three angles in the first quadrant, we can find the exact value of several other angles. Lets look at the reference angle of 45 0, or  /4.

Using Reference Angles Using the three angles in the first quadrant, we can find the exact value of several other angles. Lets look at the reference angle of 45 0, or  /4.

The Unit Circle This is the unit circle that we will be using from now on. You need to memorize this and be able to recreate it for me on every test.

Using a Reference Angle Find the exact value of the following: a)cos 4  /3 b)tan ( ) c)csc 11  /6

Using a Reference Angle Find the exact value of the following: a)cos 4  /3 b)tan ( ) c)csc 11  /6 First, we find the reference angle. Second, we determine which quadrant the angle is in. This tells us if the answer is positive or negative. Third, we find the exact value.

Using a Reference Angle Find the exact value of the following: a)cos 4  /3 b)tan ( ) c)csc 11  /6 First, we find the reference angle. Second, we determine which quadrant the angle is in. This tells us if the answer is positive or negative. Third, we find the exact value.

Using a Reference Angle Find the exact value of the following: a)cos 4  /3 b)tan ( ) c)csc 11  /6 First, we find the reference angle. Second, we determine which quadrant the angle is in. This tells us if the answer is positive or negative. Third, we find the exact value.

Using a Reference Angle You Try: Find the exact value of: a)sin 3  /4 b)cos c)tan 5  /6

Using a Reference Angle You Try: Find the exact value of: a)sin 3  /4 b)cos c)tan 5  /6

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