Download To solve a problem using trigonometric ratios

Name _______________________
Geometry 300
Date ________________________
8-4 Trigonometry Day 1
In this lesson we will look at :
- ratios of sides for different angles in right triangles
- finding a missing side using trigonometric ratios
- finding a missing angle using trigonometric ratios
Exploration #1
Label the opposite, adjacent and hypotenuse using A and D as the reference angles.
You will do one triangle, your neighbor will do the other. Share your answers at the end.
Please find the ratios of the following sides in each triangle; reduce, if necssary.
Ratio
Triangle I
𝑠𝑖𝑑𝑒 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑠𝑖𝑛𝑒 𝐴 =
𝑐𝑜𝑠𝑖𝑛𝑒 𝐴 =
Ratio
𝑠𝑖𝑑𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐴
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝐴 =
𝑠𝑖𝑑𝑒 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴
𝑠𝑖𝑑𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐴
Triangle II
𝑠𝑖𝑑𝑒 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐷
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑠𝑖𝑛𝑒 𝐷 =
𝑐𝑜𝑠𝑖𝑛𝑒 𝐷 =
𝑠𝑖𝑑𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐷
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝐷 =
What conclusion can you make about triangles ABC and DEF?
What conclusions can you make about angles A and D?
𝑠𝑖𝑑𝑒 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐷
𝑠𝑖𝑑𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐷
Exploration #2
Label the opposite, adjacent and hypotenuse using B and E as the reference angles.
Find the ratios and simplify if necessary.
Ratio
Triangle I
𝑠𝑖𝑑𝑒 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐵
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑠𝑖𝑛𝑒 𝐵 =
𝑐𝑜𝑠𝑖𝑛𝑒 𝐵 =
Ratio
𝑠𝑖𝑑𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐵
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝐵 =
𝑠𝑖𝑑𝑒 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐵
𝑠𝑖𝑑𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐵
Triangle II
𝑠𝑖𝑑𝑒 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐸
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑠𝑖𝑛𝑒 𝐸 =
𝑐𝑜𝑠𝑖𝑛𝑒 𝐸 =
𝑠𝑖𝑑𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐸
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝐸 =
What conclusion can you make about angles B and E?
What relationship is there between A and B? Between D and E?
Can you find the ratios for angle C and F? Explain.
𝑠𝑖𝑑𝑒 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐸
𝑠𝑖𝑑𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐸
A: Trigonometric Ratios:
In a ____________________ __________________, the ratios of the sides are defined as:
Sine X= __________________________________________________________
Cosine X=_________________________________________________________
Tangent X= _________________________________________________________
Practice: Finding the ratio given all sides.
Practice: Use your calculator to find the values for each angle.
1. sin 55˚
2. tan 10˚
3. cos 30˚
4. sin 12˚
5. tan 5˚
6. cos 89˚
7. sin 60˚
8. tan 60˚
9. cos 35˚
Circle the ratios that are equal. What do you notice about the angles?
B: Cofunctions
If two angles are __________________________, then
Practice: Answer in terms of cofunctions
1. cos 45o=
2. sin 30o=
3. Sin x=
C: Given a ratio, find the missing side.
5
Example: In a right triangle, tan 𝜃 = 12, find the missing side and also find cos 𝜃 , tan 𝜃.
Practice: In each of the following, find the missing side and the remaining ratios.
2
1
1. cos 𝛼 = 5
2. sin 𝛽 = 4
Why do we use trigonometry?
You want to find out the height of a ledge of a building. You do this by going outside and
measuring how far you are from the building then using a clinometer to find out the
angle measure to the ledge. Find the height using the figure below.
Step 1: Label your diagram
Step 2: Determine the ratio
and set up an equation
Step 3: Find the ratio using
the calculator, if not given.
Step 4: Cross multiply to
solve for the missing value
Practice:
Sin 38˚=
x
322
Cos 65˚=
3.5
x
Tan 42˚=
x
39
D. Given a sine, cosine or tangent ratio, find the angle measure
Using Calculator:
+
+
+
+
Example 1:
Sin x= 0.7664
Sin A=
23.5
25
cos x=0.0698
Cos x=
tan x=2.0503
15
20
Tan B=
Example 2: Find the approximate measure of ∠X.
X
8
W
Example 3: Find the missing angle. Round to the nearest tenth.
a.
12
24
b.
11
Y