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.

Trig 1: Right Triangle Trigonometry

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 h𝑙𝑒𝑛𝑔𝑡h=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=𝑠𝑥=2𝑠𝑥=2(16 )𝑥=32

Trigonometric Ratios: ratios of sides of a right triangle.

opposite

adjacent

hypotenuse

opp

adjhyp

 

 

 

3 Basic Trigonometric Ratios:3 basic:

sin

cos

tan

opphypadjhypoppadj

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

°

7cos𝜃=

𝑎𝑑𝑗h𝑦𝑝

adj

hyp

cos 35 °= 𝑥7

7 ∙cos35 °=𝑥7 ∙7

7 ∙cos35 °=𝑥 Make sure your calculator is in degrees.

𝑥=5.73406431𝑥≈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 °

Assignment:Special Right Triangles & Trig Worksheet, 1-22 all

1. Find the value of .

7th Warm-Up 3/25

tan 21°= 9𝑥

𝑥=9

tan 21 °

𝑥=23.4458

Assignment:Special Right Triangles & Trig Worksheet, 1-22 all