Warm-Up 8/13What are the six trigonometric functions and the ratios?

Q&A on yesterday’s assignment.

Rigor:You will learn the six trigonometric functions and how to use them to solve problems. Relevance:You will be able to solve real world right triangle trigonometry problems using the six trigonometric functions.

4-1a Right Triangle Trigonometry

Trigonometric Ratios: ratios of sides of a right triangle.

opposite

adjacent

hypotenuse

opp

adjhyp

 

 

 

6 Trigonometric Ratios:3 basic: 3 additional:

sin

cos

tan

opphypadjhypoppadj

csc

sec

cot

hypopphypadjadjopp

csc 𝜃=h𝑦𝑝𝑜𝑝𝑝

¿1sin 𝜃∙

1h𝑦𝑝1h𝑦𝑝

=¿1

𝑜𝑝𝑝h𝑦𝑝

csc 𝜃=1sin 𝜃

Reciprocal Functions

sec𝜃=1

𝑐𝑜𝑠𝜃

cot 𝜃=1tan 𝜃

cot 𝜃=cos𝜃sin𝜃tan𝜃=

si n 𝜃cos𝜃

2 10

θ

3

7

Example 1: Find the exact values of the 6 Trigonometric functions of

s 𝑖𝑛𝜃=𝑜𝑝𝑝h𝑦𝑝

¿ 2√107

oppadj

hyp

cos𝜃=𝑎𝑑𝑗h𝑦𝑝¿

37

ta𝑛𝜃=𝑜𝑝𝑝𝑎𝑑𝑗

¿ 2√103

cs 𝑐 𝜃=h𝑦𝑝𝑜𝑝𝑝

¿7

2√10¿ 7√1020

se𝑐 𝜃=h𝑦𝑝𝑎𝑑𝑗¿

73

𝑐𝑜𝑡 𝜃=𝑎𝑑𝑗𝑜𝑝𝑝¿

32√10

¿ 3√1020

Since any two right triangles with angle are similar, these ratios are the same, regardless of the size of the triangle.

3

4

530

40

50

Example 2: If , find the exact values of the 5 remaining trigonometric functions.

1

2√2

3

s 𝑖𝑛𝜃=13=

𝑜𝑝𝑝h𝑦𝑝

12+𝑏2=32

1+𝑏2=9𝑏2=8𝑏=√8¿ 2√2

3

¿ 12√2

=√24

¿31=3

¿ 32√2

=3√24

¿ 2√21

=2√2

cos𝜃=𝑎𝑑𝑗h𝑦𝑝

ta𝑛𝜃=𝑜𝑝𝑝𝑎𝑑𝑗

cs 𝑐 𝜃=h𝑦𝑝𝑜𝑝𝑝

se𝑐 𝜃=h𝑦𝑝𝑎𝑑𝑗

𝑐𝑜𝑡 𝜃=𝑎𝑑𝑗𝑜𝑝𝑝

Example 3: 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

not

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

not

s 𝑖𝑛𝜃=1215.7

𝜃=𝑠𝑖𝑛−1( 1215.7 )

𝜃≈50°

√−1math!

4-1a Checkpoints: TX p222, 3A; p223, 5A

Find the measure of .Find the value of .

4-1a Assignment: TX p227, 3-36 DB3