1. goal: know how to set up different trig ratios
TRIGONOMETRIC RATIOS
goal: know how to set up different trig ratios

2. Where to start… θ this is the symbol for an unknown angle measure.
It’s name is ‘Theta’.
Don’t let it scare you… it’s like ‘x’ except for angle measure… it’s a way for us to keep our variables understandable and organized.

3. Trigonometric Ratios Sine (pronounced “sign”) Cosine
Name
“say”
Sine
(pronounced “sign”)
Cosine
(pronounced “co-sign)
tangent
(pronounced “tan-gent)
Abbreviation
Abbrev.
Sin
Cos
Tan
Ratio of an angle measure
Sinθ = opposite side
hypotenuse
cosθ = adjacent side
tanθ =opposite side
adjacent side

4. hypotenuse
hypotenuse
opposite
opposite
adjacent
adjacent

5. SOH CAH TOA O S H A C H O T A One more time… Here are the ratios:
sinθ = opposite side
hypotenuse H A C
cosθ = adjacent side
hypotenuse H O T
tanθ =opposite side
adjacent side A
SOH CAH TOA

6. INE OSINE ANGENT PPOSITE DJACENT YPOTENUSE PPOSITE DJACENT YPOTENUSE
We could ask for the trig functions of the angle Θ by using the definitions.  c b
SOHCAHTOA
SOHCAHTOA Θ
INE a
OSINE
ANGENT
PPOSITE
DJACENT
YPOTENUSE
PPOSITE
DJACENT
YPOTENUSE

7. You need to pay attention to which angle you want the trig function of so you know which side is opposite that angle and which side is adjacent to it. The hypotenuse will always be the longest side and will always be opposite the right angle.
Oh, I'm acute!
This method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle.  5 4
So am I! Θ 3

8. Trigonometric Ratios
Let ∆ABC be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows.
Side adjacent to A b
cos A = =
hypotenuse c
Side opposite A a
sin A = =
hypotenuse c
Side opposite A a
tan A = =
Side adjacent to A b

9. Let’s practice… M c a N b L Write the ratio for sin L Sin L= _a c
Write the ratio for cos L
Cos L = _b_
Write the ratio for tan L
Tan L = _a_ b M c a
N b L
Let’s switch angles: Find the sin, cos and tan for Angle M:
Tan M = _b_ a
Sin M = _b_ c
Cos M = _a_ c

10. 65 a x
Practice Together:
Given each triangle, write the ratio that could be used to find x by connecting the angle and sides given.
32 b x

11. YOU DO:
56 d x
Given the triangle, write all the ratios that could be used to find x by connecting the angle and sides given. c

12. Ex. 2: Finding Trig Ratios
opposite
sin S =
hypotenuse
adjacent
cosS =
hypotenuse
opposite
tanS =
adjacent

13. Ex. 2: Finding Trig Ratios—Find the sine, the cosine, and the tangent of the indicated angle.
opposite
sin S =
hypotenuse
adjacent
cosS =
hypotenuse
opposite
tanS =
adjacent

14. Ex. 1: Finding Trig Ratios
Large Triangle
Small Triangle
opposite
sin A =
hypotenuse
adjacent
cosA =
hypotenuse
opposite
tanA =
adjacent

15. Ex. 3: Finding Trig Ratios—Find the sine, the cosine, and the tangent of 45
opposite
sin 45=
hypotenuse
adjacent
cos 45=
hypotenuse
opposite
tan 45=
adjacent √2
45

16. Ex. 4: Finding Trig Ratios—Find the sine, the cosine, and the tangent of 30
opposite
sin 30=
hypotenuse
adjacent
cos 30=
hypotenuse
opposite
tan 30=
adjacent
30 √3