MTH 112 Elementary Functions

Chapter 5 The Trigonometric Functions Section 3 –

Trigonometric Functions of Any Angle

Two Views of an Angle  Geometric   The union of two rays with a common endpoint.

Measurement is from 0  180  (limited range) to

Two Views of an Angle  Geometric  A Rotation  Position a ray on the positive x-axis with the endpoint at the origin. This is the

initial side

of the angle.

 Rotate a copy of this ray around the origin to produce the

terminal side

of the angle.

 Measurement specifies the amount & direction of rotation.

Signed Angle Measurement (Rotational View of an Angle)  Positive Angles  Counterclockwise rotation.

Signed Angle Measurement (Rotational View of an Angle)  Positive Angles  Negative Angles  Clockwise rotation.

Signed Angle Measurement (Rotational View of an Angle)  Positive Angles  Negative Angles  Right Angles      90  -90  270  -270  450   etc.

Signed Angle Measurement (Rotational View of an Angle)  Positive Angles  Negative Angles  Right Angles  Rays    0  360  -360   etc.

Signed Angle Measurement (Rotational View of an Angle)  Positive Angles  Negative Angles  Right Angles  Rays  Lines    180  -180  540   etc.

Signed Angle Measurement (Rotational View of an Angle)  How many measurements can a single angle represent?    How are these measurements related?

 n ± 360k   0  ≤ n < 360   k  non-negative integers Coterminal : Angles with the same terminal side.

Trigonometric Functions of Acute Angles of Rotation (x, y) is any point on the terminal side of the angle other than the vertex.

r



x y (x, y)

 sin  = y / r  cos   tan  = x / r = y / x

x 2 + y 2 = r 2

 sec  = r / x  csc   cot  = r / y = x / y Reference Triangle : The x-y-r right triangle.

Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle – First Quadrant

r



x y (x, y)

   sin  cos  tan  = y / r = x / r = y / x x 2 + y 2 = r 2  sec   csc  = r / x = r / y  cot  = x / y x > 0 & y > 0  All six functions are positive.

Trigonometric Functions of Any Angle – Second Quadrant

(x, y) y r x

  sin   cos   tan  = y / r = x / r = y / x x 2 + y 2 = r 2  sec   csc   cot  = r / x = r / y = x / y x < 0 & y > 0  cos  is negative & sin  is positive.

Trigonometric Functions of Any Angle – Third Quadrant  sin  = y / r  cos   tan  = x / r = y / x

(x, y) y x r

 x 2 + y 2 = r 2  sec   csc   cot  = r / x = r / y = x / y x < 0 & y < 0  cos  & sin  negative.

Trigonometric Functions of Any Angle – Fourth Quadrant  sin  = y / r  cos   tan  = x / r = y / x 

r x y (x, y)

x 2 + y 2 = r 2  sec   csc   cot  = r / x = r / y = x / y x > 0 & y < 0  cos  is positive & sin  is negative.

Summary of the Signs of the Trig Functions by Quadrant sin > 0 cos < 0 tan < 0 sin < 0 cos < 0 tan > 0 All positive sin < 0 cos > 0 tan < 0

(x, y) (x, y)

Reference Angle

( -x , y)

 

( -x , -y )



(x, y)



(x, -y )

 The first quadrant angle containing the point … ( |x|, |y| ) Where (x, y) is on the terminal side of the angle.

(x, y)

Using Reference Angles to Find Trig Values of Multiples of 30 °, 45 °, and 60° Angles.

225 ° Consider a 225 ° angle.

45 ° The reference angle is 45 °.

225 – 180 = 45 Since 225 ° is in the third quadrant … sin 45   2 2 cos 45  tan 45   2 2  1 sin 225  

The same idea can be applied to the other two quadrants.

 2 2 cos 225    tan 225   1 2 2

Given one trigonometric value of an angle and the quadrant of the terminal side of the angle, find the other five trigonometric values.

1.

2.

3.

4.

Write the given value as a fraction.

- What about negative signs?

Draw a right triangle, specify the coterminal angle (  ), and use the fraction to determine the lengths of two sides relative to the indicated angle.

Determine the other five trigonometric values using the definitions.

Determine the signs of the other five trigonometric values using the indicated quadrant.

Given one trigonometric value of an angle and the quadrant of the terminal side of the angle, find the other five trigonometric values … example!

Problem: If sec  = -2.5 and  is in the second quadrant, find sin  , cos  , & tan  .

2.5 = 5/2



5 2

sin   21 5 cos    2 5 

21

tan    21 2

Trig Values of Angles with the Terminal Side on an Axis r (0, r) Case 1: Positive y-Axis 90 ° ± 360k° (k is any non-negative integer) sin 90 ° = r / r = 1 cos 90 ° = 0 / r = 0 tan 90 ° = r / 0 ….. undefined !

(-r, 0) r Trig Values of Angles with the Terminal Side on an Axis Case 2: Negative x-Axis 180 ° ± 360k° (k is any non-negative integer) sin 180 ° = 0 / r = 0 cos 180 ° = -r / r = -1 tan 180 ° = 0 / -r = 0

Trig Values of Angles with the Terminal Side on an Axis (0, -r) r Case 3: Negative y-Axis 270 ° ± 360k° (k is any non-negative integer) sin 270 ° = -r / r = -1 cos 270 ° = 0 / r = 0 tan 270 ° = -r / 0 ….. undefined !

Trig Values of Angles with the Terminal Side on an Axis r (r, 0) Case 4: Positive x-Axis 0 ° ± 360k° (k is any non-negative integer) sin 0 ° = 0 / r = 0 cos 0 ° = r / r = 1 tan 0 ° = 0 / r = 0

Trig Values of Angles with the Terminal Side on an Axis

Angle 0 ° 90 ° 180 ° 270 ° sin

0 1 0 -1

cos

1 0 -1 0

tan

0 — 0 —

Review of Known Acute Angles

Angle 30 ° 45 ° 60 ° sin

½ 2 / 2 3 / 2

cos

3 / 2 2 / 2 ½

tan

3 / 3 1 3