Sec 2 Hon Notes – RIGHT TRIANGLE Trigonometry – 8.1
Trig Ratios: sine, cosine, tangent, cosecant, secant, cotangent
 (theta): acute angle
sin  
opposite _ leg
hypotenuse
cos  
adjacent _ leg
hypotenuse
tan  
opposite _ leg
adjacent _ leg
csc 
hypotenuse
opposite _ leg
sec  
hypotenuse
adjacent _ leg
cot  
adjacent _ leg
opposite _ leg
Memory Trick: SOH - CAH -TOA
Trig ratios only apply to right triangles!
Find the trig ratios: Find missing side lengths where necessary (reduced root form).
Give all ratios in reduced fraction form where appropriate.
1.
2.
3.
R
M
t
s
T
3
S
r
sin S =
csc S =
cos S =
sec S =
tan S =
cot S =
N
sin O =
cos M =
tan M =
4. Find the trig ratios. Remember: trig
ratios only apply to right triangles!
Hint: You will need to add a segment
to the diagram 
L
5
9
3
O
csc O =
K
P
sin K =
csc L =
cos L =
sec K =
tan K =
cot L =
sec M =
cot O =
5. For these problems you will need to draw your own
triangle. Hint: use the simplest possible side lengths
appropriate for the triangle.
Rationalize your ratios. (No roots left in denominator.)
Y
25
25
X
14
Z
cos X =
cot Z =
sec Z =
csc Z =
tan X =
sin X =
sin 45° =
csc 45° =
cos 30° =
sec 30° =
tan 60° =
cot 60° =
Calculator Trigonometry
ALWAYS CHECK the MODE
Round to 3 decimal places!!!
Finding the trig RATIOS: Why do we need a calculator for these ratios when we didn’t for the last problem set?
1.
sin 37  _______
2.
tan 53  _______
3.
sec 28  ______ (Write down the calculator steps for this.)
Finding the ANGLE!! How do you do this on the calculator?
4. If sin A = 0.4103, then A = _______
5. If cos A = 0.0912, then A = _______
6. If sec A = 1.2156, then A = _______ (Write down the calculator steps for this!!!)
Find the missing values. Show neat, complete work!!!
7. A boat travels in the following path. How far north
did it travel?
8. An architect needs to use a diagonal support in an
arch. Her company drew the following diagram.
How long does the diagonal support have to be?
9. Rennie is walking her dog. The dog’s leash is 12
10. A park has a skateboard ramp with a length of 14.2
feet long and is attached to the dog 10 feet
feet and a length along the ground of 12.9 feet. The
horizontally from Rennie’s hand, as shown in the
height is 5.9 feet. Calculate the measure of the
diagram. What is the angle formed by the leash and
angle formed by the ramp and the ground.
the horizontal at the dog's collar?
When giving an angle measure in degrees, you must have a degree symbol!!!
Sec 2 Hon Notes – RIGHT TRIANGLE Trigonometry – 8.2
(Orally review the trig ratio formulas as a class)
Angles & Special Triangles
Define the following terms:
Standard Position Angle:
Initial ray:
Terminal ray:
Coterminal angles:
Reference angles:
1st: Draw all the angles in standard position. 2nd: Find, and label the measure of, the reference angle.
1.
35º
5. -135º
8.
2. 240º
3. 300º
6. 405º
7. 137º
Give two angles coterminal with 210º _____ _____
3
4. -60º
Reference Triangles – For each angle measure below, sketch a coordinate system showing all possible angle
measures (4) with the given reference angle measure. Then construct the reference triangle for each and label the
side lengths. (Hint: The leg lengths may be negative, but the hypotenuse will always be a positive length)
Look at Geogebra – Unit Circle
30 degrees
60 degrees
45 degrees
11. Find the following trig ratios by drawing triangles (No Calculator!) on a coordinate system.
Rationalize answers.
a. sin 120°
b. cos 210°
c.
tan 330°
d. csc 135°
Sec 2 Hon Notes – RIGHT TRIANGLE Trigonometry – 8.3
Unit Circle!!
sin 𝜃 =
csc 𝜃 =
cos 𝜃 =
sec 𝜃 =
tan 𝜃 =
cot 𝜃 =
Find the value of the following:
1. cos 60° =
2. sin 120° =
3. tan 90° =
4. cot 90° =
5. sec 60° =
6. sec 60° =
7. csc −135° =
8. cos −60° =
9. tan −300° =
10. sec 0° =
11. tan 270° =
12. cot −180° =
13. sin −420° =
14. csc 540° =
15. sec −855° =
Draw triangles to find the following. No Calculator! Give all answers in reduced root/fraction form.
1. cot 300 csc300
2. cot 2 330  csc330
3. If θ is in standard position and contains the point (-8, 15), find sin θ and sec θ.
4. If 𝜃 is in standard position and contains the point (12, −14) find tan 𝜃.
5
Sec 2 Hon Notes – RIGHT TRIANGLE Trigonometry – 8.4
Radians (180º = 𝝅 𝒓𝒂𝒅𝒊𝒂𝒏𝒔) Think of radians as slicing a circle into fractions. It is very important that
you get comfortable working and thinking in radians!!!
Complete the diagram below by adding the angle measures in 𝜋 form of radians, and decimal radians.
𝜋
Example: 90° = 2 ≈ 1.57
90
0,360
180
270
Convert from degrees to radians. Write your answer in reduced fraction form in terms of π. Try to find
the value of the angle by looking at the diagram before calculating it.
1. 60º
2. -180º
3. 30º
4.
5. 135º
6. 720º
300º
Convert from radians to degrees. Remember degree marks! Try to find the value of the angle by looking
at the diagram before calculating it.
7.
2𝜋
3
8.

4
9.
11
6
10. 3𝜋
Convert from degrees to radians in decimal form (2 decimal places). Look at the diagram at the top of
this page and try to estimate the value before calculating it.
11. 23º
12. 162º
6
Convert from radians to degrees in decimal form (2 decimal places). Look at the diagram at the top of the
notes and try to estimate the value before calculating it. How do you know these angles are in radians?
13.
0.95
14. 5.32
Review: Find exact answers (trig ratios) for the following. NO Calculator. Include a labeled triangle with
your ratio.
15. sin

4
16. cos

6
17. cot
3
4
18. sec 
Review: Find the trig ratios in decimal form to two decimal places. Calculator review.
Hint: given angles are in radians. How do you know the values are angles? How do you know the angles
are in radians?
19. sin 0.67
20. csc (- 1.34)
7
Sec 2 Hon Notes – RIGHT TRIANGLE Trigonometry – 8.5 (QUIZ day)
Inverse Trig Functions: Given the ratio, find the angle measure/s
Use your unit circle to find 𝜃
1.
If sin θ =
2. If sin 𝜃 =
3.
1
2
find: a. 0    90
 3
2
find: a. 0   
If cos θ = 
2
2

2
find: a. 0    90
b. 0    360
c. no restrictions
b. 0    2
c. no restrictions
3
2
c. no restrictions
b.    
Inverse (Arc) Trig functions. Given the ratio, find the angle/s.
Arc’s have specific domains or quadrants (called principal values) where you look to find the solutions.
Trig function
Principal Values:
Arc sin or sin -1:
90    90
or
Arc cos or cos -1:
0    180
or
Arc tan or tan -1:
90    90
or


 
2
2
0  


 
2
2
Draw a triangle to find the following. Write all answers in BOTH degrees and radians.
4. Arc sin (-1/2) =
7. Sin -1
6. Arc tan 1 =
8
2
=
2
Sec 2 Hon Notes – RIGHT TRIANGLE Trigonometry – 8.6
Identities: true for all angles.
Pythagorean Identities
sin 2   cos 2   1
1  tan   sec 
2
2
1  cot 2   csc2 
Reciprocal Identities
1
sin 
1
sec  
cos 
1
cot  
tan 
csc  
Ratio Identities
Cofunction Identities
sin 
cos 
cos 
cot  
sin 


cos   sin    
2



csc   sec    
2



cot   tan    
2

tan  
Negative Identities
sin( )   sin 
cos( )  cos 
csc( )   csc 
sec( )  sec 
tan( )   tan 
cot( )   cot 


sin   cos    
2



sec   csc    
2



tan   cot    
2

Prove the following identities:
1.
sin 2  sec csc  tan 
3. Simplify
2.
cos x sin x
(sin x  1)(sin x  1)
sec2   2 tan   (1  tan  )2
**REMEMBER**
9
*Don’t invent new rules.
*Changing things to sin and cos
usually works.
*You can’t use Pythagorean unless things
are squared.
*Don’t move things across the equal sign
when proving identities.