Warm-Up 3/24-25What are three basic trigonometric functions and the their ratios?
Sine: sin
Cosine: cos
Tangent: tan
ยฟ๐๐๐h๐ฆ๐
ยฟ๐๐๐h๐ฆ๐
ยฟ๐๐๐๐๐๐
Rigor:You will learn how to solve right triangles, and find the three basic trigonometric ratios. Relevance:You will be able to solve real world problems using trigonometric ratios.
Special Right Triangles45แต- 45แต- 90แต: both legs are congruent and the length of the hypotenuse is times the length of a leg.
30แต- 60แต- 90แต: The length of the hypotenuse is 2 times the shorter leg and the other leg is times the shorter leg.
๐ =๐๐๐ h๐๐๐๐๐ก
h=h๐ฆ๐๐๐ก๐๐๐ข๐ ๐=๐ โ2
๐ = h๐ ๐๐๐ก ๐๐๐h=h๐ฆ๐๐๐ก๐๐๐ข๐ ๐=2๐ ๐=๐๐๐๐๐๐๐=๐ โ3
60แต
30แต
x
16 345แต
x
x
Example 1: Solve the triangles.
a. b.
12=๐ฅ โ212
โ2=๐ฅ
โ2โ2โ
12โ22
=๐ฅ
6 โ2=๐ฅ
s
16โ3=๐ โ316=๐
๐ฅ=2๐ ๐ฅ=2(16 )๐ฅ=32
Since any two right triangles with angle are similar, side ratios are the same, regardless of the size of the triangle.
3
4
530
40
50
2 10
ฮธ
3
7
Example 2: Find the exact values of the 3 basic Trigonometric functions of
s ๐๐๐=๐๐๐h๐ฆ๐
ยฟ 2โ107
oppadj
hyp
cos๐=๐๐๐h๐ฆ๐
ยฟ37
ta๐๐=๐๐๐๐๐๐
ยฟ 2โ103
Example 3: If , find the exact values of the 2 remaining basic trigonometric functions.
1
2โ2
3
s ๐๐๐=13=๐๐๐h๐ฆ๐
12+๐2=32
1+๐2=9๐2=8๐=โ8ยฟ 2โ2
3
ยฟ 12โ2
=โ24
cos๐=๐๐๐h๐ฆ๐
ta๐๐=๐๐๐๐๐๐
Example 4: Find the value of . Round to the nearest tenth, if necessary.
x
ยฐ
7
cos๐=๐๐๐h๐ฆ๐
adj
hyp
cos 35 ยฐ=๐ฅ7
7 โcos35 ยฐ=๐ฅ7โ7
7 โcos35 ยฐ=๐ฅ Make sure your calculator is in degrees.
๐ฅ=5.73406431
๐ฅโ5.7
Example 5: Use a trigonometric function to find the measure of . Round to the nearest degree.
1215.7
opp
hyp
๐=49.84753016 ยฐ
๐
s ๐๐๐=๐๐๐h๐ฆ๐
s ๐๐๐=1215.7
๐=๐ ๐๐โ1( 1215.7 )
๐โ50ยฐ
Checkpoints:
3. Find the measure of .2. Find the value of .
1. Fill out chart with exact values.
12โ32
โ33
12
โ32
โ3
โ22โ22
1
sin 53 ยฐ=15๐ฅ
๐ฅ=15
sin 53 ยฐ
๐ฅ=18.7820
cos๐=512
๐=cosโ 1( 512 )๐=65 ยฐ