Download Math 10C Ch. 2 Review notes

Math 10C Ch. 2 Review Notes
All of our triangles must have 90° angles. All the angles in a triangle add to 180°.
Make sure your calculator is in degree mode.
The two acute angles add to 90° (∠𝐴 + ∠𝐵 = 90°)
B
A
C
Angles are written in capital letters. Sides are written in lowercase letters. Angles and
sides are named across from each other.
B
c
a
A
b
C
Pythagorean Theorem:
𝑎2 + 𝑏 2 = 𝑐 2
Example:
B
15.6
11.3
A
C
Since we have the hypotenuse, use backwards pythagorus.
15.62 − 11.32 = 𝑏 2
10.8 = 𝑏
We can actually use the “word” SOHCAHTOA to help us remember the ratios:
S:
O:
H:
C:
A:
H:
T:
O:
A:
Sine
opposite
hypotenuse
Cosine
adjacent
hypotenuse
Tangent
opposite
adjacent
When we have the angle, we use the normal sin/cos/tan buttons.
When we want the angle, we use the 2nd sin/cos/tan buttons.
Sine Ratio
𝑠𝑖𝑛𝐴 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Example #1:
E
31
22
F
D
To find ∠𝐹, use the sine ratio.
𝑠𝑖𝑛𝐹 =
22
31
22
∠𝐹 = 𝑠𝑖𝑛−1 (31) = 45°
Example #2:
A
B
21°
b
17.3
C
To find side b, use the sine ratio.
𝑠𝑖𝑛21° =
𝑏
17.3
𝑏 = 17.3𝑠𝑖𝑛21° = 6.3
When the unknown is on the top, multiply.
Cosine Ratio
𝑐𝑜𝑠𝐴 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Example #1
S
11.8
R
5.3
To find ∠𝑅, use the cosine ratio.
𝑐𝑜𝑠𝑅 =
5.3
11.8
5.3
∠𝐹 = 𝑐𝑜𝑠 −1 (11.8) = 63°
T
Example #2:
H
I
14
20°
G
To find side h, use the cosine ratio.
𝑐𝑜𝑠20° =
14
ℎ
14
ℎ = 𝑐𝑜𝑠20° = 14.9
**when the unknown is on the bottom, divide.
Tangent Ratio
𝑡𝑎𝑛𝐴 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
Example #1
C
17.1
B
A
12.3
To find ∠𝐴, use the tangent ratio.
𝑡𝑎𝑛𝐴 =
17.1
12.3
17.1
∠𝐴 = 𝑡𝑎𝑛−1 (12.3) = 54°
Example #2:
4.8
T
R
r
21°
S
To find side r, use the tangent ratio.
𝑡𝑎𝑛21° =
4.8
𝑟
4.8
𝑟 = 𝑡𝑎𝑛21° = 12.5
Clinometers
Regular clinometers measure the angle of elevation (bottom angle)
Drinking straw (homemade) clinometers measure the top angle.
x
22
1.2 m
20 m
We would use the tangent ratio to find the x value.
𝑥
𝑡𝑎𝑛22° = 20 = 20𝑡𝑎𝑛22° = 8.0 …
Then we need to add the height of the person holding the clinometer to get the actual
height (make sure you keep all the decimals) 8.0 … + 1.2 = 9.3 𝑚
Solving triangles
To solve a triangle means to find all the sides and angles. There are two types of
triangles: given 2 sides or given 1 side and 1 angle.
Example #1: (given 2 sides)
Solve triangle ABC.
A
17
B
12
C
Answer: we are missing ∠𝐴, ∠𝐵 𝑎𝑛𝑑 𝑠𝑖𝑑𝑒 𝑏.
172 − 122 = 𝑏 2
12 = 𝑏
12
𝑠𝑖𝑛𝐴 = 17
12
𝑠𝑖𝑛−1 (17) = 45°
∠𝐵 = 90 – 45 = 45°.
Example #2: (given a side and an angle)
Solve this triangle. Give the measures to the nearest tenth where necessary.
D
C
12.1°
15.4
E
Missing ∠𝐷 and side c and side e.
So, ∠𝐷 = 90 – 12.1 = 77.9°.
𝑐𝑜𝑠12.1 =
15.4
𝑒
15.4
𝑒 = 𝑐𝑜𝑠12.1 = 15.7
𝑐
𝑡𝑎𝑛12.1 = 15.4
𝑑 = 15.4𝑡𝑎𝑛12.1 = 3.3
Two Triangle Problems
Remember: angle of elevation (up from the horizontal). Angle of depression (down
from the horizontal)
1. Two triangles attached to each other.
Example #1: Find ∠𝑋𝑌𝑍
Y
4 cm
X
12.1°
3 cm
3
𝑡𝑎𝑛−1 ( ) = 36.8 …
4
6
𝑡𝑎𝑛−1 ( ) = 56.3 …
4
36.8 … + 56.3 … = 93°
6 cm
Z
2. One triangle is inside another triangle.
Example #2: Find ∠𝑋𝑌𝑍
T
8
15 cm
Y
17 cm
X
6
Z
15
) = 28.0 …
17
14
∠𝑇𝑌𝑍 = 𝑡𝑎𝑛−1 ( ) = 43.0 …
15
∠𝑇𝑌𝑋 = 𝑐𝑜𝑠 −1 (
∠𝑋𝑌𝑍 = 43.0 … − 28.0 … = 15.0°