1. 5.2 Trigonometric Ratios in Right Triangles
What are the six trigonometric ratios in a right triangle?
How do we find the values of the six trig ratios?

2. What is a trig ratio?
A trigonometric ratio is a ratio of two sides of a right triangle in relation to a specific acute angle. Each trig ration has a specific name to distinguish what sides are being used. 𝒔𝒊𝒏 𝒏𝒂𝒎𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒕𝒓𝒊𝒈 𝒓𝒂𝒕𝒊𝒐 𝜶 𝒂𝒏𝒈𝒍𝒆 𝒃𝒆𝒊𝒏𝒈 𝒓𝒆𝒇𝒆𝒓𝒆𝒏𝒄𝒆𝒅 = 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒓𝒂𝒕𝒊𝒐 𝒐𝒇 𝒔𝒊𝒅𝒆𝒔 sin 𝜶= 𝟑 𝟓 is read as “the sine of alpha is 3/5” -NOTE 𝒂 is read “the square root of a”, you have to tell what you are taking the square root OF, just like you have to tell what ANGLE you are using for the ratio.
Too often the students write just sin, they need to understand that 𝛼 is a variable and can change. This is especially important when we go to create the graph of sine/cos/tan. They also need to understand that sin is an instruction to do something just like a square is a instructions to do something.

3. What do we need to know about trig ratios?
The trig ratios are instructions on how to find a particular number, in order to do that you need to know some background information.
Pythagorean theorem
𝑎 2 + 𝑏 2 = 𝑐 2
a, b are legs of a right triangle (across from acute angles)
c is the hypotenuse (longest side, across from right angle)
Opposite/Adjacent/Hypotenuse
The side “opposite” depends on the angle, if you change the angle you change the side that is “opposite”, similarly with “adjacent.”
What side is “opposite” to C?

4. Example 1: Find the missing side of each triangle
𝑎=𝑏=4
𝑎=8, 𝑏=15
= 𝑐 2
= 𝑐 2
16+16=32= 𝑐 2
𝑐 2 =289
𝑐= 32 = 16∙2 =4 2
𝑐=17
𝑎=9, 𝑐=41
𝑎=3, 𝑐=5
9 2 + 𝑏 2 =4 1 2
3 2 + 𝑏 2 = 5 2
𝑏 2 =1600
𝑏 2 =16
𝑏=40
𝑏=4
𝑎=𝑥, 𝑏=𝑥 3
𝑏=5, 𝑐=10
𝑥 2 + 𝑥 = 𝑐 2
𝑎 =1 0 2
𝑥 2 +3 𝑥 2 = 𝑐 2
𝑎 2 =75
4 𝑥 2 = 𝑐 2
𝑎= 75 = 25∙3 =5 3
𝑐= 4 𝑥 2 =2𝑥

5. Example 2: Identify the opposite, adjacent, and hypotenuse to the given angle
Angle A
Opposite = 4
Adjacent = 3
Hypotenuse = 5
Angle C
Angle B
Not well defined ??
We often tell students in geometry that we cant use the right angle, which confuses students when we go to do the unit circle since there is a sin/cos/ tan value for the unit circle. I tell them that we CAN use the 90 because there is a clear opposite, it just the same as the hypotenuse which makes a ratio of 1 for sin. The way I explain the cos is that if we were to draw a triangle with a hypotenuse of 1 and an opposite side of REALLY close to 1, then the third side (adjacent) must have to be REALLY small, if we were to make the opposite side closer and closer to 1, the adjacent side would be closer and closer to 0 so when we go to find cos 90 = adjacent/hyp = 0/1=0.

6. What are the six trig ratios?
Original Trig Ratios
Words
Symbols
Ratio
Sine of 𝜃
sin 𝜃
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Cosine of 𝜃
cos 𝜃
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Tangent of 𝜃
tan 𝜃
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
Reciprocal Trig Ratios
Words
Symbols
Reciprocal
Ratio
Cosecant of 𝜃
csc 𝜃
1 sin 𝜃
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
Secant of 𝜃
sec 𝜃
1 cos 𝜃
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
Cotangent of 𝜃
cot 𝜃
1 tan 𝜃
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

7. Example 3: given one trig ratio, find the other 5 for the given angle
Note: in order to find all the trig ratios we must first know ALL the sides and then be able to identify the opposite, adjacent and hypotenuse. It is often helpful to sketch a picture.
sec 𝜷 = 𝟓 𝟒
sin 𝜷 =
cos 𝜷 =
tan 𝜷 =
csc 𝜷 =
cot 𝜷 = H 5 O 3 𝛽
3 5 4 A
4 5
Find all the sides
𝑎 = 5 2
𝑎 2 =9
𝑎=3
Identify O, A, H
3 4
5 3
4 3

8. Example 3: given one trig ratio, find the other 5 for the given angle
Note: in order to find all the trig ratios we must first know ALL the sides and then be able to identify the opposite, adjacent and hypotenuse. It is often helpful to sketch a picture.
B. cos 𝜷 = 8 17
sin 𝜷 =
tan 𝜷 =
csc 𝜷 =
sec 𝜷 =
cot 𝜷 = 𝛽 H 17
15 17 8 A
15 8 15 O
17 15
Find all the sides
𝑎 =
𝑎 2 =225
𝑎=15
Identify O, A, H
17 8
8 15

9. Example 4: Find the trig ratios for the special right triangles
𝟑𝟎°
sin 𝟑𝟎° =
cos 𝟑𝟎° =
tan 𝟑𝟎° =
csc 𝟑𝟎° =
sec 𝟑𝟎° =
cot 𝟑𝟎° = H
𝑥 2𝑥 = 1 2 O
𝑥 3 2𝑥 =
𝑥 𝑥 3 = = A
2𝑥 𝑥 =2
Identify O, A, H
Does it matter what we call A or O?
2𝑥 𝑥 3 = =
𝑥 3 𝑥 = = 3

10. Example 4: Find the trig ratios for the special right triangles
B. 𝟒𝟓°
sin 𝟒𝟓° =
cos 𝟒𝟓° =
tan 𝟒𝟓° =
csc 𝟒𝟓° =
sec 𝟒𝟓° =
cot 𝟒𝟓° = H
𝑥 𝑥 2 = = A
𝑥 𝑥 2 = = O
𝑥 𝑥 =1
𝑥 2 𝑥 = 2
Identify O, A, H
Does it matter what we call A or O?
𝑥 2 𝑥 = 2
𝑥 𝑥 =1

11. Example 4: Find the trig ratios for the special right triangles
sin 𝟔𝟎° =
cos 𝟔𝟎° =
tan 𝟔𝟎° =
csc 𝟔𝟎° =
sec 𝟔𝟎° =
cot 𝟔𝟎° = H
𝑥 3 2𝑥 = A
𝑥 2𝑥 = 1 2
𝑥 3 𝑥 = = 3 O
2𝑥 𝑥 3 = =
Identify O, A, H
Does it matter what we call A or O?
2𝑥 𝑥 =2
𝑥 𝑥 3 = =

12. Summary
Find the six trig ratios (of 𝜶) for a triangle with the side opposite 𝜶 measuring 6, and the hypotenuse of the triangle measuring 12.
If tan 𝜷 = 𝟑 𝟒 , find sin 𝜷 .
Given triangle ABC, with right angle B, which trig ratio could be used to find the length of 𝑨𝑪 ?
cos 𝐴 = 𝐵𝐶 𝐴𝐶
sin 𝐴 = 𝐵𝐶 𝐴𝐶
tan 𝐶 = 𝐵𝐶 𝐴𝐶
cos 𝐵 = 𝐵𝐶 𝐴𝐶

13. Summary
Find the six trig ratios (of 𝜶) for a triangle with the side opposite 𝜶 measuring 6, and the hypotenuse of the triangle measuring 12.
Using Pythagorean theorem the last side is which makes this a 30°−60°−90° triangle so see example 4C.
If tan 𝜷 = 𝟑 𝟒 , find sin 𝜷 .
Find the 3rd side using Pythagorean theorem, which will be 5. See example 3A.
Given triangle ABC, with right angle B, which trig ratio could be used to find the length of 𝑨𝑪 ?
cos 𝐴 = 𝐵𝐶 𝐴𝐶
𝐬𝐢𝐧 𝑨 = 𝑩𝑪 𝑨𝑪
tan 𝐶 = 𝐵𝐶 𝐴𝐶
cos 𝐵 = 𝐵𝐶 𝐴𝐶
DRAW A PICTURE A B C