Station 1: Coterminal Angles
We learned at the start of class that coterminal angles are angles which share the same initial
and terminal side. For example, 90°, 450°, and - 270° are coterminal.
Generally, it is easier to graph an angle when it is expressed from 0° to 360°. So, we use
coterminal angles to simplify graphing.
If we have a positive angle greater than 360°, we can find its smaller coterminal angle by
subtracting 360° until we reach an angle between 0° and 360°.
Ex: Given 1040°, find the coterminal angle between 0° and 360°.
1040° - 360° = 680°
680° - 360° = 320°
If we have a negative angle, we can find its positive coterminal angle by adding 360° until we
reach an angle between 0° and 360°.
Ex: Given -225°, find the coterminal angle between 0° and 360°.
-225° + 360° = 135°
If we start with any angle, we can find a positive coterminal angle by adding 360°. We can also
find a negative coterminal angle by subtracting 360°.
Ex: Given 45°, find a positive and a negative coterminal angle.
Positive: 45° + 360° = 405°
Negative: 45° - 360° = -315°
Complete these exercises after reading the information on the first page.
#1-4: Given the following angles, find a coterminal angle between 0° and 360°.
1.
2.
3.
4.
-72°
445°
-290°
1200°
#5-8: Given the following angles, find a positive and negative coterminal angle.
5.
6.
7.
8.
16°
250°
-640°
765°
Station 2: Reference Angles
We learned at the start of class that a reference angle is the acute angle formed by the
terminal side of an angle and the x-axis. For example, 45° is the reference angle of 135°.
Remember:
- 0° - 90° in Quadrant I
- 90° - 180° in Quadrant II
- 180° - 270° in Quadrant III
- 270° - 360° in Quadrant IV
The first step to finding a reference angle is to figure out which quadrant the angle is currently
in. If the angle is very large or is negative, sometimes it can be helpful to first find the coterminal
angle between 0° and 360° before trying to find the reference angle.
If the angle lies in:
- Quadrant I: the angle is its own reference angle
- Quadrant II: subtract the angle measure from 180° (180° - angle)
- Quadrant III: subtract 180° from the angle measure (angle - 180°)
- Quadrant IV: subtract the angle measure from 360° (360° - angle)
Ex: Find the reference angle of 200°.
200° is between 180° and 270°, so it lies in Quadrant III. Therefore, I will subtract 180°
from the angle measure.
200° - 180° = 20°
Ex: Find the reference angle of -192°.
First, I will find the coterminal angle of -192°.
-192° + 360° = 168°
168° is between 90° and 180°, so it lies in Quadrant II. Therefore, I will subtract the
angle measure from 180°.
180° - 168° = 12°
If we have a reference angle, we can also find an angle in any quadrant with that reference angle.
If we want an angle in:
- Quadrant I: the reference angle is the angle
- Quadrant II: subtract the reference angle from 180° (180° - RA)
- Quadrant III: add 180° to the reference angle (180° + RA)
- Quadrant IV: subtract the reference angle from 360° (360° - RA)
Ex: A reference angle measures 35°. Find an angle in each quadrant with that reference angle.
Quadrant I: the reference angle is the angle. 35°
Quadrant II: subtract the RA from 180°.
180° - 35° = 145°
Quadrant III: add the RA to 180°.
180° + 35° = 215°
Quadrant IV: subtract the RA from 360°
360° - 35° = 325°
Complete these exercises after reading the information.
#1-4: Find the reference angle of the following angles in standard position.
1.
2.
3.
4.
112°
548°
-50°
303°
#5-6: Given the following reference angles, find an angle in each quadrant with that
reference angle.
5. 75°
6. 22°
Station 3: Complementary and Supplementary Angles
We learned at the start of class that complementary angles add up to 90° and supplementary
angles add up to 180°.
To find an angle’s complement, subtract the angle from 90°
Ex: Find the complement of 35°.
90° - 35° = 55°
To find an angle’s supplement, subtract the angle from 180°.
Ex: Find the supplement of 35°.
180° - 35° = 145°
You can only find complements for positive angles less than 90°. You can only find supplements
for positive angles less than 180°.
Complete these exercises after reading the information above.
#1-8: Find the complement and supplement of the angle, if possible.
1.
2.
3.
4.
5.
6.
7.
8.
30°
122°
88°
100°
210°
-42°
21°
65°
Station 4: Radian Measure
We learned at the start of class that radians are another unit of measure for angles. We know
that 360° = 2π radians. We can use that conversion factor to convert between degrees and
radians. (However, understand that when we write an angle measured in radians, we do not use
the word “radians.”)
Think about how you convert from minutes to seconds. You know 1 minute = 60 seconds. So,
you can convert 2.25 minutes to seconds by multiplying by the conversion factor:
60 seconds
2.25 minutes ∙
= 𝟏𝟑𝟓 𝐬𝐞𝐜𝐨𝐧𝐝𝐬
1 minute
Notice, we put minutes in the denominator of the conversion factor so that minutes would cancel
out, leaving us with seconds. Similarly, we can convert 240 seconds to minutes by multiplying
by the conversion factor:
1 minute
240 seconds ∙
60 seconds
= 𝟒 𝐦𝐢𝐧𝐮𝐭𝐞𝐬
This time, we put seconds in the denominator of the conversion factor so that seconds would
cancel out, leaving us with minutes. Notice the 1 always goes with the minute and the 60 always
goes with the seconds, but we can switch which goes in the numerator and denominator of the
fraction. Now, since we know 360° = 2π, we can use that as a conversion factor.
Ex: Convert 120° to radians.
We want to convert from degrees to radians, so we need degrees to cancel. We will put
degrees in the denominator of the conversion factor.
120° ∙
Ex: Convert
2𝜋
360°
3𝜋
4
=
240𝜋
360
=
𝟐𝝅
𝟑
to degrees.
We want to convert from radians to degrees, so we need our 𝜋s to cancel. This means
degrees goes in the numerator of the conversion factor.
3𝜋
4
∙
360°
2𝜋
=
1080°
8
= 𝟏𝟑𝟓°
Complete these exercises after reading the information on the first page.
#1-4: Convert the following angles from degrees to radians. Find exact values.
1.
2.
3.
4.
30°
540°
-135°
240°
#5-8: Convert the following angles from radians to degrees. Find exact values.
2𝜋
5. 3
6. 𝜋
11𝜋
7. 6
𝜋
8. − 4
Station 5: Evaluating Trigonometric Functions
You may recall from last week that given a point on the coordinate plane, we can find an angle
and the corresponding trigonometric functions. We discovered the following about the quadrants:
We can use the phrase “All Students Take Classes” to remember that ALL functions are positive
in Quadrant I, SINE is positive in Quadrant II, TANGENT is positive in Quadrant III, and
COSINE is positive in Quadrant IV.
Similarly, the reciprocal functions are positive in the same quadrants where their parent functions
are positive. So, secant is positive when cosine is positive, etc.
With this information, if we have only one trigonometric function and know what quadrant it is
in, we should be able to determine the values of the other five trigonometric functions. Let’s try
an example.
Here is the information I am given:
5
sin(θ) = 13 and tan(θ) < 0
My first step is to find the length of the opposite, adjacent, and hypotenuse sides.
opposite
5
Since sin(θ) = hypotenuse and sin(θ) = 13 , I know:
opposite: 5
hypotenuse: 13
I can find the remaining side using the Pythagorean Theorem:
𝑎2 + 𝑏 2 = 𝑐 2
52 + adjacent 2 = 132
25 + adjacent 2 = 169
adjacent 2 =144
adjacent = 12
Now I have all three of my sides:
opposite: 5
adjacent: 12
hypotenuse: 13
Now, I need to think about which quadrant I am in.
The problem told me that sine is positive but tangent is negative. That means I am in Quadrant II.
Finally, I can write my six ratios.
I plug in my opposite, adjacent and hypotenuse sides in the appropriate places, and remember to
make tangent, cosine, cotangent, and secant negative because I am in the second quadrant.
5
12
5
sin(θ) = 13, cos(θ) = − 13, tan(θ) = − 12, csc(θ) =
13
5
13
, sec(θ) = − 12, cot(θ) = −
Complete these exercises after reading the information.
#1-4: Find the values of the six trigonometric functions of θ given the following
information.
8
1. cos(θ) = − 10 𝐚𝐧𝐝 tan(θ) > 0
15
2. sin(θ) = − 17 𝐚𝐧𝐝 tan(θ) < 0
2
3. tan(θ) = 3 𝐚𝐧𝐝 csc(θ) > 0
5
4. sec(θ) = 4 𝐚𝐧𝐝 cot(θ) < 0
12
5