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Warm-Up 3/24-25What are three basic trigonometric functions and the their ratios?
Sine: sin
Cosine: cos
Tangent: tan
¿𝑜𝑝𝑝h𝑦𝑝
¿𝑎𝑑𝑗h𝑦𝑝
¿𝑜𝑝𝑝𝑎𝑑𝑗
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 Triangles45ᵒ- 45ᵒ- 90ᵒ: both legs are congruent and the length of the hypotenuse is times the length of a leg.
30ᵒ- 60ᵒ- 90ᵒ: The length of the hypotenuse is 2 times the shorter leg and the other leg is times the shorter leg.
𝑠=𝑙𝑒𝑔 h𝑙𝑒𝑛𝑔𝑡
h=h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒=𝑠 √2
𝑠= h𝑠 𝑜𝑟𝑡 𝑙𝑒𝑔h=h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒=2𝑠𝑙=𝑙𝑜𝑛𝑔𝑙𝑒𝑔=𝑠√3
60ᵒ
30ᵒ
x
16 345ᵒ
x
x
Example 1: Solve the triangles.
a. b.
12=𝑥 √212
√2=𝑥
√2√2∙
12√22
=𝑥
6 √2=𝑥
s
16√3=𝑠√316=𝑠
𝑥=2𝑠𝑥=2(16 )𝑥=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
θ
3
7
Example 2: Find the exact values of the 3 basic Trigonometric functions of
s 𝑖𝑛𝜃=𝑜𝑝𝑝h𝑦𝑝
¿ 2√107
oppadj
hyp
cos𝜃=𝑎𝑑𝑗h𝑦𝑝
¿37
ta𝑛𝜃=𝑜𝑝𝑝𝑎𝑑𝑗
¿ 2√103
Example 3: If , find the exact values of the 2 remaining basic trigonometric functions.
1
2√2
3
s 𝑖𝑛𝜃=13=𝑜𝑝𝑝h𝑦𝑝
12+𝑏2=32
1+𝑏2=9𝑏2=8𝑏=√8¿ 2√2
3
¿ 12√2
=√24
cos𝜃=𝑎𝑑𝑗h𝑦𝑝
ta𝑛𝜃=𝑜𝑝𝑝𝑎𝑑𝑗
Example 4: Find the value of . Round to the nearest tenth, if necessary.
x
°
7
cos𝜃=𝑎𝑑𝑗h𝑦𝑝
adj
hyp
cos 35 °=𝑥7
7 ∙cos35 °=𝑥7∙7
7 ∙cos35 °=𝑥 Make sure your calculator is in degrees.
𝑥=5.73406431
𝑥≈5.7
Example 5: Use a trigonometric function to find the measure of . Round to the nearest degree.
1215.7
opp
hyp
𝜃=49.84753016 °
𝜃
s 𝑖𝑛𝜃=𝑜𝑝𝑝h𝑦𝑝
s 𝑖𝑛𝜃=1215.7
𝜃=𝑠𝑖𝑛−1( 1215.7 )
𝜃≈50°
Checkpoints:
3. Find the measure of .2. Find the value of .
1. Fill out chart with exact values.
12√32
√33
12
√32
√3
√22√22
1
sin 53 °=15𝑥
𝑥=15
sin 53 °
𝑥=18.7820
cos𝜃=512
𝜃=cos− 1( 512 )𝜃=65 °