1. 9.1 – Trigonometric Ratios (PART 1)

2. Why is Trigonometry important?
Flight of Planes
Sound Waves (study in music theory classes)
Current of the Ocean
Architecture
Navigation/Surveying
Without Trig. we would have NEVER made it to the moon!
Launch and Orbital Trajectories

3. Topic One Solving for missing pieces of information.

4. Right Triangle Trig. Functions
Let  be an acute angle of a right triangle.
sin 𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
cos 𝜃 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
tan 𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
SOH
CAH
TOA

5. Identifying the correct angle makes a difference!

6. Evaluate the Trig. Functions for 𝜃
sin 𝜃 = 15 𝜃
hypotenuse
sin 𝜃 = 15 17 17 8
cos 𝜃 =
adjacent
cos 𝜃 = 8 17
tan 𝜃 =
opposite
ta𝑛 𝜃 = 15 8

7. Evaluate the Trig. Functions for 𝜃
hypotenuse
sin 𝜃 = 14 𝜃
sin 𝜃 =
5 29 23
opposite
cos 𝜃 =
cos 𝜃 =
tan 𝜃 =
adjacent
ta𝑛 𝜃 = 23 14

8. THIS IS WHEN THE MODE OF YOU CALCULATOR MATTERS!!!!
Find the Value of x
THIS IS WHEN THE MODE OF YOU CALCULATOR MATTERS!!!!
When you evaluate trig. functions at angles you need to be in the correct angle units.
sin 47 = 𝑥 15 15 x
47
hypotenuse
15 sin 47 =𝑥
𝟏𝟎.𝟗𝟕=𝒙
opposite

9. Find the Value of x cos 31 = 15 𝑥 x cos 31 =15 x= 15 cos 31 𝒙=𝟏𝟕.𝟓𝟎 x
31
hypotenuse
x cos 31 =15
x= 15 cos 31
adjacent
𝒙=𝟏𝟕.𝟓𝟎

10. Topic Two Solving for ALL of the missing pieces of information.

11. Solving Right Triangles
When asked to SOLVE A RIGHT TRIANGLE
Find ALL missing sides
Find ALL missing angles
What to Use when solving a right triangle:
SOH – CAH – TOA
Pythagorean Thm.
Triangle Sum Thm.
Inverse Trig (solve for unknown angles)

12. Pythagorean Thm. 𝑙𝑒𝑔 2 + 𝑙𝑒𝑔 2 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 2 Triangle Sum Theorem All angles in a triangle add = 180∘

13. Change your calc to radian mode and see what happens…
Inverse Trig – to find UNKNOWN angles!!!
𝜃= sin −1 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝜃= cos −1 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝜃= tan −1 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 15 𝜃 18
𝜃= tan −
𝜃= tan −1 1.2
𝜽=𝟓𝟎.𝟏𝟗∘
Change your calc to radian mode and see what happens…
𝜃= tan −1 1.2
𝜽=.𝟖𝟕𝟔

14. Solve the Right Triangle
55∘
10.65
𝒄=𝟕.𝟒𝟔
Side c:
sin 35 = 𝑐 13 OR
𝑐 2 = 13 2
𝑐 2 =169
𝑐 2 =55.58
𝒄=𝟕.𝟒𝟔
Side a:
cos 35 = 𝑎 13
13cos 35 =𝑎
𝒂=𝟏𝟎.𝟔𝟓
𝑚∠𝐴=180−90−35
𝒎∠𝑨=𝟓𝟓∘

15. Solve the Right Triangle
𝟐𝟐.𝟔𝟐∘
𝟔𝟕.𝟑𝟖∘ 𝟏𝟐
Find side e:
(Pythagorean triple!)
5 2 + 𝑒 2 = 13 2
25+ 𝑒 2 =169
𝑒 2 =144
e=𝟏𝟐
sin ∠𝐷 = 5 13
∠𝐷= sin −
∠𝑫=𝟐𝟐.𝟔𝟐∘
𝑚∠𝐸=180−90−22.62
𝒎∠𝑬=𝟔𝟕.𝟑𝟖∘

16. Topic Three Angles of Elevation and Depression

17. Angle of Elevation & Depression

18. Angle of Elevation
An ant looks up at the top of a tree with an angle of elevation of 62∘. (The ants height is negligible). If the tree is 15 feet tall, how far away is the ant from the top of the tree?

19. Angle of Depression
From the top of a vertical cliff 40 m high, the angle of depression of an object that is level with the base of the cliff is 34º. How far is the object from the base of the cliff?

20. Homework Topic One: Textbook Pg. 333#5, 6
Topic Two: Textbook Pg. 334 #1, 2, 8
Topic Three: Workbook Pg. 67 #16-18