1. Warm-Up #32 Tuesday, 5/10/2016
Solve for x and find all of the missing angles.
In triangle JKL, JK=15, JM = 5, LK = 13, and PK = 9. Determine whether line segment JL is parallel to line segment MP. Justify your answer.

2. Homework
Area of Regular Polygons_ finish

3. Basic Trigonometry

4. The Trigonometric Functions
SINE
COSINE
TANGENT

5. Represents an unknown angle
Greek Letter q
Prounounced “theta”
Represents an unknown angle

6. Right Triangle Trigonometry
The hypotenuse is the longest side and is always opposite the right angle.
The opposite and adjacent sides refer to another angle, other than the 90o. A

7. Parts of a Right Triangle
The hypotenuse will always be the longest side, and opposite from the right angle.
Imagine that you are at Angle A looking into the triangle.
The opposite side is the side that is on the opposite side of the triangle from Angle A.
The adjacent side is the side next to Angle A.

8. Parts of a Right Triangle
Now imagine that you move from Angle A to Angle B.
From Angle B the opposite side is the side that is on the opposite side of the triangle.
From Angle B the adjacent side is the side next to Angle B.

9. hypotenuse
hypotenuse
opposite
opposite
adjacent
adjacent

10. Trig Ratios Each of the 3 ratios has a name
Hypotenuse
Each of the 3 ratios has a name
The names also refer to an angle
Opposite A
Adjacent

11. Trig Ratios If the angle changes from A to B
Hypotenuse
If the angle changes from A to B
Opposite A
The way the ratios are made is the same
Adjacent

12. SOHCAHTOA SOHCAHTOA is pronounced “Sew Caw Toe A” and it means B
Hypotenuse
Here is a way to remember how to make the 3 basic Trig Ratios
Opposite A
Adjacent
1) Identify the Opposite and Adjacent sides for the appropriate angle
SOHCAHTOA is pronounced “Sew Caw Toe A” and it means
Sin is Opposite over Hypotenuse, Cos is Adjacent over Hypotenuse, and Tan is Opposite over Adjacent
Put the underlined letters to make
SOH-CAH-TOA

13. Examples of Trig Ratios
First we will find the Sine, Cosine and
Tangent ratios for Angle P. 20 12
Adjacent
Next we will find the Sine, Cosine, and
Tangent ratios for Angle Q Q 16
Opposite
Remember SohCahToa

14. Similar Triangles and Trig Ratios
Triangles are similar if the ratios of the lengths of the corresponding side are the same.
Triangles are similar if they have the same angles
All similar triangles have the same trig ratios for corresponding angles

15. Finding an angle from a triangle
To find a missing angle from a right-angled triangle we need to know two of the sides of the triangle.
We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio.
14 cm
6 cm C 1.
Find angle C
Identify/label the names of the sides.
b) Choose the ratio that contains BOTH of the letters.

16. 14 cm
6 cm C 1.
We have been given the adjacent and hypotenuse so we use COSINE:
Cos A = h a
Cos A =
Cos C =
Cos C =
C = cos-1 (0.4286)
C = 64.6o

17. Find angle x 2.
8 cm
3 cm x
Given adj and opp
need to use tan:
Tan A = a o
Tan A =
Tan x =
Tan x =
x = tan-1 (2.6667)
x = 69.4o

18. 3.
12 cm
10 cm y
Given opp and hyp
need to use sin:
Sin A =
sin A =
sin x =
sin x =
x = sin-1 (0.8333)
x = 56.4o

19. Finding a side from a triangle
7 cm k
30o 4.
We have been given the adj and hyp so we use COSINE:
Cos A =
Cos A =
Cos 30 =
Cos 30 x 7 = k
6.1 cm = k

20. 4 cm r
50o 5.
We have been given the opp and adj so we use TAN:
Tan A =
Tan A =
Tan 50 =
Tan 50 x 4 = r
4.8 cm = r

21. 12 cm k
25o 6.
We have been given the opp and hyp so we use SINE:
Sin A =
sin A =
sin 25 =
Sin 25 x 12 = k
5.1 cm = k

22. x = x
5 cm
30o 1.
Cos A =
Cos 30 =
x = cm
4 cm r
50o 2.
Tan 50 x 4 = r
4.8 cm = r
Tan A =
Tan 50 = 3.
12 cm
10 cm y
y = sin-1 (0.8333)
y = 56.4o
sin A =
sin y =
sin y =

23. Applications Involving Right Triangles
A surveyor is standing 115 feet from the base of the Washington Monument. The surveyor measures the angle of elevation to the top of the monument as 78.3. How tall is the Washington Monument?
Solution:
where x = 115 and y is the height of the monument. So, the height of the Washington Monument is
y = x tan 78.3
 115( )  555 feet.

24. Ex.
A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree?
tan 71.5° ?
tan 71.5°
71.5°
y = 50 (tan 71.5°) 50
y = 50 ( )

25. Ex. 5
A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge?
cos 60°
x (cos 60°) = 200
200
60° x x
X = 400 yards