Trig Functions of Any Angle

Trig Functions of Any AngleSection 4.4In first section, we calculated trig functions for acute angles.

In this section, we are going to extend these basic definitions to cover any angle.

Plot the point (-3,4)

Label the hypotenuse r and find its length.

r

= 5 5 -3 4Sin =Cos =Tan =

Definitions of Trig Functions of Any AngleLet be an angle in standard position with (x,y) a point on the terminal side. Then:

Csc =

Sec =

Cot = Sin =

Cos =

Tan =

Find the 6 trig functions of given that the ray ends at the point (-15, -8)

-15 -8 17

Csc =

Sec =

Cot = Sin =

Cos =

Tan =

Find the 6 trig functions of given that the ray ends at the point (12, -5)

12 -5 13

Csc =

Sec =

Cot = Sin =

Cos =

Tan =

QuadrantsIn which quadrants was the Sine positive?I and II

In which quadrants was the Cosine positive?I and IV

In which quadrants was the Tangent positive?I and IIIQuadrants

All Trig Functionsare positiveSine is positiveCosine is positiveTangent is positiveAllStudentsTakeCalculusWhat quadrant is in if:

Sin > 0 and Cos < 0

Tan > 0 and Cos < 0

Sin < 0 and Tan < 0

Cos > 0 and Tan > 0 II III IV IGiven that Tan = - and Sin > 0, find the remaining 5 trig functions of .

What quadrant?II-24725

Csc =

Sec =

Cot = Sin =

Cos =

Tan =

Given that Cos = - and Sin < 0, find the remaining 5 trig functions of .

What quadrant?III-45-3

Csc =

Sec =

Cot = Sin =

Cos =

Tan =

Given that Sin = - and Tan < 0, find the remaining 5 trig functions of .

What quadrant?IV-15178

Csc =

Sec =

Cot = Sin =

Cos =

Tan =

What did we learnHow to find the trig functions of an angle given a point on its terminal side

How to determine the quadrant of an angle based on trig functions

How to find the trig functions based on one function and criteria

Homework: Page 297, 1-24 oddFind the Sin, Cos, and Tan trig functions of given that the ray ends at the point (5,0)

5 y = 0

Sin =

Cos =

Tan = Quadrant AnglesOn our Cartesian plane, we have 5 critical points:

Find the Sine of these 5 anglesSin 0 = 0 Sin = 1

Sin = 0 Sin = -1

Sin 2 = 0 Graph of the Sine CurveUsing these 5 points, we can create the Sine Curve

0

Quadrant AnglesUsing the same process, find the Cos of the 5 critical points.Cos 0 = 1 Cos = 0

Cos = -1 Cos = 0

Cos 2 = 1 Graph of the Cosine CurveUsing these 5 points, we can create the Sine Curve

0

Reference AnglesThe acute angle formed by the terminal side of an angle and the horizontal axis.

For an angle , we use to denote the reference angleReference AnglesWhat is the reference angle for 210

Where is there an acute angle between the terminal side of the angle and the horizontal axis?

= 210 180

= 30Reference AnglesFind the reference angles for the following:

330

225

-225

750

= 360 - 330= 30= 225 - 180= 45= -180 - -225= 45= 750 - 720= 30Reference AnglesIn general, for any angle

= = 180 - = - = - 180 = - = 360 - = 2 - Reference AnglesFind the reference angle for

2nd Quadrant: =

=

Reference AnglesSo far, all we have been finding are reference angles.

We use reference angles to find the exact value of angles that are not acute.

We will use this for the remainder of the year.

GTK Good to KnowFinding the Exact ValueFind the reference angle

Find the trig function of the reference angle

Check the sign of the functionSin 2001. Find the reference angle

Find the Sin of the reference angle

3. Is it positive or negative?

Cos 3301. Find the reference angle

2. Find the Sin of the reference angle

3. Is it positive or negative?

Find the Sin, Cos, and Tan of 135Reference Angle = Quadrant =

Sin 135 =

Cos 135 =

Tan 135 =

Find the Sin, Cos, and Tan of -240Reference Angle = Quadrant =

Sin -240 =

Cos -240 =

Tan -240 =

Find the Sin, Cos, and Tan ofReference Angle = Quadrant =

Sin =

Cos =

Tan =

Find the:Sin

Csc

Tan

Csc

Cot