MATH 31 LESSONS Chapters 6 & 7: Trigonometry 1. Trigonometry Basics

Section 6.1: Functions of Related Values Read Textbook pp. 250 - 267

A. Standard Position Angles are created by rotating an “arm” from the positive x-axis, which is called the initial arm. Where the angle ends is called the terminal arm. Counterclockwise angles are positive. Clockwise angles are negative.

e.g. Sketch the angle +200 in standard position. y x

y x start Angles in standard position are measured from the positive x-axis, which is the initial arm.

90 y 180 0, 360 x start 270 It is useful to include the major angles at each axis. The clockwise direction is positive.

90 y 200 180 0, 360 x start 270 Now, we sweep the arm 200 counterclockwise, since it is a positive angle.

e.g. Sketch the angle 310 in standard position. y x

y x start Angles in standard position are measured from the positive x-axis, which is the initial arm.

y x start 0, 360 270 180 90 It is useful to include the major angles at each axis. This time, we want the negative angles (clockwise).

Now, we sweep the arm 310 clockwise, since it is a negative angle. y x start 0, 360 270 180 90 310 Now, we sweep the arm 310 clockwise, since it is a negative angle.

Radians Another measure of angle, apart from degrees, is radians. To convert from radians to degrees (and vice versa), use the following information: 2  radians = 360 or,  radians = 180

y  0, 2 x These are the major angles at each axis in radians.

Converting from Radians to Degrees e.g. Convert radians to degrees.

Since  radians = 180, you can simply substitute 180 wherever you see .

Converting from Radians to Degrees e.g. Convert 252 to radians.

degrees goes on the bottom, since it must cancel Recall,  radians = 180

B. Trig Ratios When we define the trigonometric ratios, we will use a circle (rather than a triangle). In this way, we can deal with angles that are bigger than 180 (as well as negative angles).

Primary Trig Ratios Consider a circle of radius r. y x r

We consider any point P that is on the circumference of the circle. Its general coordinates will be (x, y). P (x, y)

We can create a triangle with height y and base x. The hypotenuse will be r, since it represents the radius of the circle. r y x P (x, y) 

Using the Pythagorean theorem, r y x P (x, y)  r is the radius of the circle, so it must always be positive (r > 0)

We can now use “Soh Cah Toa” to define each primary trig ratio. r y x P (x, y) 

“Soh Cah Toa” P (x, y) r y  x

“Soh Cah Toa” P (x, y) r y  x

“Soh Cah Toa” P (x, y) r y  x

Reciprocal Trig Ratios

Ex. 1 Evaluate cos  and csc  if Answer in exact values. Do not find the angle. Try this example on your own first. Then, check out the solution.

 Determine the quadrant of the angle y x  x The angle is in quadrant 2, where x is negative and y is positive

 Sketch the triangle Thus, x = -12 and y = 5 Remember, x is negative and y is positive

r x = -12 y = 5 

 Find r r x = -12 y = 5 

 Find cos  r = 13 x = -12 y = 5 

 Find csc  r = 13 x = -12 y = 5 

C. Reference and Coterminal Angles Reference Angle The reference angle is the acute angle (< 90) between the terminal arm and the nearest x-axis. Reference angles are always positive.

e.g. What is the reference angle for 260?

First, we sketch the angle. y x start 0, 360 90 180 270 260 First, we sketch the angle.

The reference angle is the angle between the terminal arm and the nearest x-axis. y x 180 80 270 The reference angle is 80

Coterminal Angles Two angles that have the same terminal arm are called coterminal angles.

y x  y x   and  are coterminal.

For any given reference angle (e.g. 50), Note: For any given reference angle (e.g. 50), there are an infinite number of coterminal angles. y 50 x

For any given reference angle (e.g. 50), Note: For any given reference angle (e.g. 50), there are an infinite number of coterminal angles. y x 50

The smallest positive angle to the terminal arm (130) is called the principal angle. y 130 50 x

The next positive angle with the same terminal arm is constructed by adding 360 to the principal angle. y 130 + 360 = 490 50 x

The first negative angle with the same terminal arm is constructed by subtracting 360 from the principal angle. y 50 x 130 - 360 = -230

In general, we can find all coterminal angles by adding or subtracting multiples of 360 from the principal angle. i.e. If 1 and 2 are coterminal, then 2 = 1 + (360) n , where n  I n belongs to the integers. Thus, n  ... , -3, -2, -1, 0, 1, 2, 3, ...

If 1 and 2 are coterminal, then 2 = 1 + (360) n , where n  I or in radian form, 2 = 1 + 2 n , where n  I

D. Exact Trig Values (Using Special Triangles) It is crucial that you remember the exact values for the trig ratios of the following angles:  30  45  60 To do so, we need to use special triangles.

60 "Soh Cah Toa" 60 2 1 sin 60 = 2 cos 60 = 1 2 tan 60 = 1

30 "Soh Cah Toa" 30 2 1 sin 30 = 1 2 cos 30 = 2 tan 30 = 1

45 "Soh Cah Toa" 45 1 sin 45 = 1 cos 45 = 1 tan 45 = 1

E. CAST This is a simple but effective way to remember the signs of all trig ratios in each quadrant. C A S T 1 2 3 4

sin + cos + tan + S All + 2 1 3 4 T C

C A Sine + T 1 2 3 4 sin + cos  tan 

C A S Tan + 1 2 3 4 sin  cos  tan +

Cos + A S T 1 2 3 4 sin  cos + tan 

If we define the trig circle with a radius of 1 unit, F. Unit Circle If we define the trig circle with a radius of 1 unit, called the unit circle, then finding exact values for the trig ratios is much more straightforward. 1 1

“Soh Cah Toa” 1 y = sin   x The sine ratio is simply the y-coordinate.

“Soh Cah Toa” 1 y  x = cos  The cosine ratio is simply the x-coordinate.

“Soh Cah Toa” P (x, y) r y  x The tangent ratio remains y over x.

In general, P (sin , cos ) r y = sin   x = cos 

Building the unit circle ... (1, 0) (0, 1) (-1, 0) (0, -1) Start with the axes.

Next, add the special triangle ratios in quadrant 1. (1, 0) (0, 1) (-1, 0) (0, -1) Next, add the special triangle ratios in quadrant 1. Remember, x = cos  and y = sin 

It is crucial that you memorize this special triangle. (1, 0) (0, 1) (-1, 0) (0, -1) It is crucial that you memorize this special triangle.

Ex. 2 Evaluate the following exactly (without your calculator): Try this example on your own first. Then, check out the solution.

 Convert to degrees (if you need to) Recall,  radians = 180

 Evaluate the special angles using the unit circle (1, 0) (0, 1) (-1, 0)

Recall, sine is the y-coordinate (1, 0) (0, 1) (-1, 0) 90 Recall, sine is the y-coordinate

Recall, cosine is the x-coordinate (1, 0) (0, 1) (-1, 0) 180 Recall, cosine is the x-coordinate

(1, 0) (0, 1) (-1, 0) 360 Recall, tangent is y / x

 Answer the question:

Ex. 3 Evaluate the following exactly (without your calculator): Try this example on your own first. Then, check out the solution.

 Convert to degrees (if you need to) Recall,  radians = 180

Note that sin 2  = (sin ) 2

 Evaluate the special angles using the unit circle (1, 0) (0, 1) (-1, 0) These special angles can be read off the unit circle directly.

 Answer the question:

Ex. 4 Express the following as a function of its related acute angle and then evaluate: Try this example on your own first. Then, check out the solution.

 First, sketch the angle 90 y 120 180 0 x start

 Next, find the reference the angle y 120 60 x The reference angle is 60

 Using CAST, determine whether the trig ratio is positive or negative y S A 60 x T C Since the angle is in quadrant 2, sine is positive.

 Express the trig ratio in terms of the reference angle:

to evaluate the special angle exactly:  Use the unit circle to evaluate the special angle exactly: (1, 0) (0, 1) (-1, 0) Recall, sine is the y-coordinate.

Ex. 5 Express the following as a function of its related acute angle and then evaluate: Try this example on your own first. Then, check out the solution.

 First, convert to degrees and a primary trig ratio: Recall,  radians = 180

 Next, sketch the angle 90 y 330 180 0, 360 x start 270

 Next, find the reference the angle y 330 x 30 The reference angle is 30

 Using CAST, determine whether the trig ratio is positive or negative y A S x 30 T C Since the angle is in quadrant 4, tangent is negative (only cosine is positive)

 Express the trig ratio in terms of the reference angle:

to evaluate the special angle exactly:  Use the unit circle to evaluate the special angle exactly: (1, 0) (0, 1) (-1, 0) Recall, tangent is y / x.

 Answer the question: