The Fundamental Identity

andReference Angles

Now to discover my favorite trig identity, let's start with a right triangle and the Pythagorean Theorem.

Rewrite trading terms places

a

b c

222 cba 222 cab Divide all terms by c2

c2 c2 c2

1 22

ca

cb Move the exponents to the outside

Look at the triangle and the angle and determine which trig function these are.

oh

This one is sin

a

h

This one is cos

1 cossin 22

THE FUNDAMENTAL IDENTITY

1 cossin 22 This is a short-hand way you can write trig functions that are squared

There is another method for finding the other 5 trig functions of an acute angle when you know one function. This method is to use fundamental identities.

31sin

We'd still get csc by taking reciprocal of sin

csc 3

Now use my favorite trig identity1cossin 22 Sub in the value of sine that you know

1cos31 2

2

Solve this for cos

98cos2

322

98cos

This matches the answer we got with the other method

You can easily find sec by taking reciprocal of cos.

We won't worry about because angle not negative

square root both sides

Let's list what we have so far:31sin

csc 3

We need to get tangent using fundamental identities.

cossintan

Simplify by inverting and multiplying

322

31

tan

322cos

Finally you can find cot by taking the reciprocal of this answer.

223sec

223

31

221

22cot

TRIGONOMETRY FUNCTIONSOF 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 and is denoted ’ (read theta prime).

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.

referenceangle ’

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

45° =135°

-1

12

1 2sin13522

oh

2

221135cos

ha

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.

‘

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?

30°

=210°

210°-180°=30°

The reference angle is the amount past 180° of

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.

2-13

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.

210cscYou should be thinking csc is the reciprocal of sin and sin is opposite over hypotenuse so csc is hypotenuse over opposite.

21

2

210tan33

31

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

13h13

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

cottanseccoscscsin

‘

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.

1213

oh

135

ha

125

oa

1213

oh

513

ah

125

oa

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

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 ho

111

90cos ha

010

90tan ao

01dividing by 0 is

undefined so the tangent of 90° is undefined

90csc 111

90sec undef01

90cot 010

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

sin ho

010

cos ha

111

tan ao

010

csc undef01

sec 11

1

cot undef01

Remember x is adjacent, y is opposite

and hypotenuse

here is 1

Table 4 on page 528 in your book lists the trig values of the quadrantal angles. You'll need to know them but don't memorize the table, learn how to draw the picture like we did here to find them.

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.

Let’s try finding a couple of reference angles in radians.

HINT: Since it is 2 all the way around a circle, half way around (a straight line) is .

What is the measure of this reference angle?

= reference

angle ’

34

34 4

‘

Let’s try another one.

Find the reference angle for

First draw a picture and label (We know that will be in Quadrant III)

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?

=

The reference angle is the amount past of

76

76

76

76 6

6