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Warm-Up 8/13. What are the six trigonometric functions and the ratios?. Q&A on yesterday’s assignment. - PowerPoint PPT Presentation
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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