I Want to Talk About the Basis of Trigonometry, the Unit Circle

MA 15400Lesson 4Section 6.3

Trigonometric Values of Real Numbers

I want to talk about the basis of trigonometry, the unit circle. Draw a coordinate plane and only mark one unit in each direction.. Draw a circle that is centered at the origin and has a radius of one. This is the unit circle. Mark you x-axis and y-axis and also write cosine next to the x and sine next the y.

Draw a right triangle with a central angle  = 60,

mark point P on the circle. The x-value of the

coordinate is equal to the cos(60) and the

y-value of the coordinate is equal to sin(60)

cos(60) = and the sin(60) =

If we draw the corresponding right triangle for a 45 central angle, the corresponding point is, therefore cos(45) = and sin(45) =

Draw the corresponding right triangle for a 30 central angle and the corresponding point is , therefore cos(30) = and the sin(30) =

Since tangent = then; tan(60) = ,

tan(30)= and, tan(45)=

From the unit circles shown with angles of 30°, 45°, and 60° in standard position, you can see that angles less than 45° have an x-coordinate for the point on the unit circle greater than the y-coordinate. (The cosine θ is larger than the sine θ.) Angles greater than 45° have an x-coordinate for the point on the unit circle less than the y-coordinate. (The cosine θ is smaller than the sine θ.) For the 45° angle, the sine and cosine values are equal. If you keep these pictures in mind, it will help you ‘memorize’ and distinguish between the sine and cosine values for the 30 and 60 degrees.

What do the pictures say about the tangent values? Since , the following must be true.

Find the exact value of x and y.

We can pick points on the unit circle that are not in QI

and find the sine and cosine of the angle.

Find the exact value of the six trigonometric functions:

 = 90 =  = 0

Review:

cos  = , tan  < 0csc  = , sec  < 0

Use your calculator to find the following:

cos(33) and cos(– 33) sin(18) and sin(– 18) tan(63) and tan(–63)

FORMULAS FOR NEGATIVES

Therefore:

cos(–) = cos()sin(–) = – sin()tan(-) = – tan()

A point P(x, y) is on the unit circle. Find the value of the six trig functions.

Let P(t) be a point on the unit circle. Find:

P(t + ), P(t – ), P(–t), P(–t – )

Compare P(t) with P(-t). Notice the cosines are the same and the sines are the opposites. This reinforces what the formulas for negatives stated.

Let P be a point on the unit circle. Find the exact values of the six trig functions.

 (This would be a 45° angle in QII.)

Use the formula for negative to find the exact value.

(Think: 45°)

Verify the identity:

csc(–x)cos(–x)=–cot(x)