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Trigonometry Review Chapter 4
Radian: the measure of the central angle (∠PCA) of a circle subtended by an arc
that is the same length as the radius of the circle. In other words, when the
radius and arc length are equal, you get one radian. This brings us the formula:
a
 .
r
1
1 radian is about 6 th of a circle (57.3 degrees)
One full rotation of a circle is 360˚ or 2π radians. One-half rotation is 180˚ or π
radians.
𝜋
𝜋
One-quarter rotation is 90˚ or 2 radians. One-eighth rotation is 45˚ or 4 radians
Angle measures without units are considered to be in radians.
Conversion Chart
Degrees to Radians multiply by

180
Radians to Degrees multiply by
180

Angle in standard position: It has its centre at the origin and its initial arm along
the positive x-axis. The terminal arm then rotates about the origin. Angles in a
counterclockwise direction are positive and angles in a clockwise direction are
negative.
Coterminal angle: angles that have the same terminal arm (end at the same
place). To find these, either add or subtract 360˚ OR add or subtract 2 
depending on whether your question is in degrees or radians.
Principal angle: the angle that is between 0 and 360˚ or between 0 and 2  .
General Form:
By adding or subtracting multiples of one full rotation, you can write an infinite
number of angles that are coterminal with any given angle.
𝜃 ± (360°)𝑛 𝑜𝑟 𝜃 ± 2𝜋𝑛, n E N
Arc Length:
a  R
where a= arc length
R = radius
 = angle in radians
A unit circle is a circle with a radius of 1 unit:
y (0,1)
(-1,0)
(1,0)
0
x
(0,-1)
The equation of the unit circle is 𝑥² + 𝑦² = 1
We can use our unit circle or special triangles to find sinA, cosA, tanA of special
  
angles: 30, 45, 60˚ in degrees or , , in radians.
6 4 3
√2
Make sure you know the unit circle and special triangles in both degrees and
radians.
We can also use the special triangles or unit circle to find ratios of angles that are
rotated about the circle. But remember: If the question is on the arm of the axes,
use your arm to punch the calculator because the answer will always be 1, -1,
0 or undefined.
You can define sine, cosine and tangent with SOHCAHTOA or with x, y and r.
With a unit circle with radius of 1:
𝑥
cos 𝜃 = = 𝑥 , 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒
1
𝑦
sin 𝜃 = = 𝑦 , 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒
1
Therefore, you can describe the coordinates of any point 𝑃(𝜃) as (cos 𝜃, sin 𝜃)
Reference Angle: the acute angle whose vertex is the origin and whose arms
are the terminal arm of the angle and the x-axis.
CAST Rule:
Sine ratio
positive
All ratios
positive
Tangent
ratio
positive
Cosine ratio
positive
Reciprocal Trigonometric Ratios:
Each primary trig function has its corresponding reciprocal function. These are
on the formula sheet.
cosecant ratio  csc 
1
sin 
secant ratio  sec 
cotangent ratio  cot  
1
tan 
1
cos
Quadrant Rules:
Q1:
Q2:
Q3:
Q4:
Reference Angle
180º – r.a. or 𝜋 − 𝑟. 𝑎.
180º + r.a. or 𝜋 + 𝑟. 𝑎.
360º – r.a. or 2𝜋 − 𝑟. 𝑎.
The trigonometric ratios of any angle can be written as the same function of a
positive acute angle called the reference angle with the sign of the ratio being
determined by the CAST rule.
i.e. cos120º = -cos60º
Here is an example where you need to determine the exact value of an angle that
is not in the first quadrant.
Ex. sin 225
**if you know the unit circle, you can just read the answer off the circle.
**if you’re using the special triangles, use the following steps:
1. What quadrant is the angle in? 3
2. What is the reference angle?  225 180  45
1
2
4. Use the CAST rule to determine the pos/neg sign: it will be negative
1
5. Answer: 2
If you are looking for the tangent of an angle and you are using the unit circle,
sin q
remember that tanq =
. Since the denominators will be the same, simply
cosq
divide the y numerator by the x numerator to get tangent.
3. Find the sin45º from the special triangles. sin 45 
You can determine approximate values for sine, cosine, and tangent using a
calculator. If the question says that you are allowed to round, you may use the
calculator because calculators can determine trigonometric values for angles
measured in degrees or radians. You will need to set the mode to the correct
angle measure. On the graphing calculator, it is under MODE.
Remember: Normal trig button when you have the angle, second trig button
when you want the angle.
The inverse calculator keys return one answer only, when there are often two
angles with the same trigonometric function value in any full rotation. In general,
it is best to use the reference angle applied to the appropriate quadrants
containing the terminal arm of the angle.
The notation [0, 𝜋] represents the interval from 0 to π inclusive and is another
way of writing 0 ≤ 𝜃 ≤ 𝜋.
To solve first degree equations,
1. Get the trig ratio on one side and the number on the other
2. Determine the quadrants that your answer is in based on the domain
given.
**If you can use the unit circle, just read the answers in the correct
quadrants. If you can’t continue with the following steps or use the special
triangles.
3. Find the reference angle.
4. Find your answers using the quadrant rules.
Ex. 5 sin 𝜃 + 2 = 1 + 3𝑠𝑖𝑛𝜃 , 0 ≤ 𝜃 < 2𝜋 (this solution uses the special
triangles)
1
2 sin 𝜃 = −1, 𝑡𝑜 𝑠𝑜𝑙𝑣𝑒: sin 𝜃 = − 2
𝑠𝑖𝑛−1 (0.5) = 30° 𝑜𝑟
𝜋
6
Sin is negative in quadrants III and IV
𝜋
Quad III: 𝜋 + 6 =
𝜋
7𝜋
6
Quad IV: 2𝜋 − 6 =
11𝜋
6
To solve second degree equations,
1. Get everything on one side.
2. Either factor the quadratic equation or use the quadratic formula.
3. Set each answer equal to zero and solve.
4. Find the reference angle for each.
5. Determine the quadrants that your answer is in based on the domain
given.
6. Find your answers using the quadrant rules.
Ex. tan²θ – 5tanθ + 4 = 0, 0 ≤ 𝜃 < 2𝜋
Need to factor: (tanθ – 1)(tanθ – 4) and then need to solve for each
factor:
tan 𝜃 = 1, 𝑡𝑎𝑛−1 (1) = 45° 𝑜𝑟
𝜋
4
𝜋 5𝜋
Positive in QI and QIII: 4 ,
4
Second factor: tan 𝜃 = 4, 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 𝑚𝑜𝑑𝑒 𝑡𝑎𝑛−1 (4) = 1.33
Postive in QI, and QIII: 1.33 and 4.467 (π + 1.33)
Make sure you can also write the general solution answer. You need to write it
with multiples of the period.
Ex. Solve cscq = 2 and give the general solution
First, cscq is the reciprocal of sinθ. Therefore, sin q =
1
. It is positive in QI and
2
QII. Using either the unit circle or the special triangles, our angles will be
p
6
and
5p
. To find the general solution, we will add one full rotation. Our two answers
6
p
5p
are:
+ 2np , ne I and
+ 2np , ne I .
6
6