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Trigonometric Function of General Angles

credit: Shawna Haider

TRIGONO

METRY FUNCTIO

NSOF GENERAL ANGLES

Our method of using right triangles only works for acute angles. Now we will see how we can find the trig function values of any angle. To do this we'll place angles on a rectangular coordinate system with the initial side on the positive x-axis.

HINT: Since it is 360° all the way around a circle, half way around (a straight line) is 180°

If is 135°, we can find the angle formed by the negative x-axis and the terminal side of the angle. This is an acute angle and is called the reference angle.

referenceangle

What is the measure of this reference angle?

=135°

180°- 135° = 45°

Let's make a right triangle by drawing a line perpendicular to the x-axis joining the terminal side of the angle and the x-axis.

Let's label the sides of the triangle according to a 45-45-90 triangle. (The sides might be multiples of these lengths but looking as a ratio that won't matter so will work)

45° =135°

The values of the trig functions of angles and their reference angles are the same except possibly they may differ by a negative sign. Putting the negative on the 1 will take care of this problem.

-1

12

211

Now we are ready to find the 6 trig functions of 135°

This is a Quadrant II angle. When you label the sides if you include any signs on them thinking of x & y in that quadrant, it will keep the signs straight on the trig functions. x values are negative in quadrant II so put a negative on the 1

-145°

=135°12

1 2sin135

22

o

h

Notice the -1 instead of 1 since the terminal side of the angle is in quadrant II where x values are negative.

11

10135tan

a

We are going to use this method to find angles that are non acute, finding an acute reference angle, making a triangle and seeing which quadrant we are in to help with the signs.

2

2

2

1135cos

h

a

Let denote a nonacute angle that lies in a quadrant. The acute angle formed by the terminal side of and either the positive x-axis or the negative x-axis is called the reference angle for .

Let's use this idea to find the 6 trig functions for 210°First draw a picture and label (We know that 210° will be in Quadrant III)

Now drop a perpendicular line from the terminal side of the angle to the x-axis

The reference angle will be the angle formed by the terminal side of the angle and the x-axis. Can you figure out it's measure?

=210°

210°-180°=30°

The reference angle is the amount past 180° of

30°

Label the sides of the 30-60-90 triangle and include any negative signs depending on if x or y values are negative in the quadrant.

2-1

3

30°

210°

2-1

3

You will never put a negative on the hypotenuse. Sides of triangles are not negative but we put the negative sign there to get the signs correct on the trig functions.

210csc

You should be thinking csc is the reciprocal of sin and sin is opposite over hypotenuse so csc is hypotenuse over opposite.

21

2

210tan3

3

3

1

210cos2

3

Using this same triangle idea, if we are given a point on the terminal side of a triangle we can figure out the 6 trig functions of the angle.

Given that the point (5, -12) is on the terminal side of an angle , find the exact value of each of the 6 trig functions.

First draw a picture

(5, -12)

Now drop a perpendicular line from the terminal side to the x-axis

Label the sides of the triangle including any negatives. You know the two legs because they are the x and y values of the point

5

-12

Use the Pythagorean theorem to find the hypotenuse

222 125 h

13h

13

Given that the point (5, -12) is on the terminal side of an angle , find the exact value of each of the 6 trig functions.

(5, -12)

5

-1213

cottan

seccos

cscsin

We'll call the reference angle . The trig functions of are the same as except they possibly have a negative sign. Labeling the sides of triangles with negatives takes care of this problem.

12

13

o

h

13

5

h

a

12

5

o

a

12

13

o

h

5

13

a

h

12

5

o

a

The Signs of Trigonometric Functions

Since the radius is always positive (r > 0), the signs of the trig

functions are dependent upon the signs of x and y.

Therefore, we can determine the sign of the functions by

knowing the quadrant in which the terminal side of the angle

lies.

In quadrant I both the x and y values are positive so all trig functions will be positive

+

+

All trig functions positive

In quadrant II x is negative and y is positive.

_+

We can see from this that any value that requires the adjacent side will then have a negative sign on it.

Let's look at the signs of sine, cosine and tangent in the other quadrants. Reciprocal functions will have the same sign as the original since "flipping" a fraction over doesn't change its sign.

sin is +cos is -tan is -

__

In quadrant IV, x is positive and y is negative .

_+

So any functions using opposite will be negative.

Hypotenuse is always positive so if we have either adjacent or opposite with hypotenuse we'll get a negative. If we have both opposite and adjacent the negatives will cancel

sin is -cos is +tan is -

In quadrant III, x is negative and y is negative.

sin is -cos is -tan is +

All trig functions positive

sin is +cos is -tan is -

sin is -cos is +tan is -

sin is -cos is -tan is +

To help remember these sign we look at what trig functions

are positive in

each quadrant.

AS

T C

Here is a mnemonic to help you remember.

(start in Quad I and go counterclockwise)

AllStudents

Take Calculus

To find the sine, cosine, tangent, etc. of angles whose terminal side

falls on one of the axes , we will use the

unit circle.

3(..., , , 0, , , , 2 ,...)

2 2 2

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

0

2

3

2

Unit Circle:

Center (0, 0)

radius = 1

x2 + y2 = 1

Trigonometric Functions Quadrantal Angle

What about quadrantal angles?

We can take a point on the terminal side of quadrantal angles and use the x and y values as adjacent and opposite respectively. We use the x or y value that is not zero as the hypotenuse as well.

Try this with 90°

(0, 1)

We can take a point on the terminal side of quadrantal angles and use the x and y values as adjacent and opposite respectively. We use the x or y value that is not zero as the hypotenuse as well (but never with a negative).

90sin h

o1

1

1

90cos h

a0

1

0

90tan a

o

0

1dividing by 0 is undefined so the tangent of 90° is undefined

90cosec 11

1

90sec undef0

1

90cot 01

0

Let's find the trig functions of (-1, 0)

sin h

o0

1

0

cos h

a1

1

1

tan a

o0

1

0

cosec undef0

1

sec 11

1

cot undef0

1

Remember x is adjacent, y is opposite

and hypotenuse

here is 1

Coterminal angles are angles that have the same terminal side.

62°, 422° and -298° are all coterminal because graphed, they'd all look the same and have the same terminal side.

62°422°-298°

Since the terminal side is the same, all of the trig functions would be the same so it's easiest to convert to the smallest positive coterminal angle and compute trig functions.

Reference Angles

for The reference angles in Quadrants II, III, and IV.

′ = – (radians) ′ = 180 – (degrees)

′ = – (radians) ′ = – 180 (degrees)

′ = 2 – (radians) ′ = 360 – (degrees)

Finding Reference Angles

angle ′.Find the reference

a. = 300

b. = 2.3

c. = –135

Trigonometric Functions of Real Numbers

Trigonometric Functions of Any Angle

Definitions of Trigonometric Functions of Any Angle

Let be an angle in standard position with (x, y) a point on the

terminal side of and 2 2r x y

sin csc

cos sec

tan cot

y r

r y

x r

r xy x

x y

y

x

(x, y)

r

Evaluate each trigonometric function using Reference Angle.

a. cos

b. tan(–210 )

c. csc

Evaluating Trigonometric Functions

(a) – Solution

Because = 4 / 3 liesis

in Quadrant III, the reference angle

As shown in Figure (a).

Moreover, the cosine is negativeQuadrant III, so

in Figure (a)

(b) – Solution

Because –210 + 360 = 150 , it follows that –210 is150 .coterminal with the second-quadrant angle

is ′ = 180 – 150 = 30 ,So, the reference angle as shownin Figure (b)

Figure (b)

cont’d

Finally, because the tangent ishave

negative in Quadrant II, you

tan(–210 ) =

=

(–) tan 30

.

(b) – Solution

(c) – Solution

Because (11 / 4) – 2 = 3 / 4, it follows

that 11 / 4 iscoterminal with the second-quadrant angle 3 / 4.

–is ′ = (3 / 4) / 4,So, the reference angle = as shownin Figure (c)

Figure (c)

cont’d

Because the cosecant is positive in Quadrant II, you have

(c) – Solution

Evaluating Trigonometric Functions

of .Let (–3, 4) be a point on the terminal.

side Find thesine, cosine, and tangent of

Solution:

x = –3, y = 4,

Solution

Try this:

13-5

y

x

(-12, -5)

-12

sin

cos

tan

csc

sec

cot

y

rx

ry

xr

y

r

xx

y

Given and , find the values of the five

other trig function of .

Next Problems...

8cos

17 cot 0

Given and , find the values of the five

other trig functions of .

3cot

8 2

Find the value of the six trig functions for

For Quadrantal Angles

2

sin2

cos2

tan2

1csc

2

1sec

2

cot2

y

x

y

x

y

x

x

y

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

0

3

2

2

Find the value of the six trig functions for

For Quadrantal Angles

7

sin 7

cos 7

tan 7

1csc 7

1sec 7

cot 7

y

x

y

x

y

xx

y

Find one positive and one negative coterminal angle of

For Coterminal Angles

3

4

Using Coterminal Angle to Find the Exact Value of a Trigonometric Funcion

Using reference angles and the special reference triangles, we can find the exact values of the common angles.

To find the value of a trig function for any common angle

1. Determine the quadrant in which the angle lies.

2. Determine the reference angle.

3. Use one of the special triangles to determine the function value for the reference angle.

4. Depending upon the quadrant in which lies, use the appropriate sign (+ or –).

Trig Functions of Common Angles

Examples

Give the exact value of the trig function (without using a calculator).

1. 2.

5sin

6

3cos

4

Trig Functions of “Uncommon” Angles

To find the value of the trig functions of angles that do NOT reference 30°, 45°, or 60°, and are not quadrantal, we will use the calculator. Round your answer to 4 decimal places, if necessary.

Make sure the Mode setting is set to the correct form of the angle: Radian or Degree

To find the trig functions of csc, sec, and cot, use the reciprocal identities.

Examples

Evaluate the trig functions to four decimal places.

1. 2.

sec( 2.5) csc 23 38' 45"

Online Notes and Presentations

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