Transcript Holt McDougal Algebra 2

Right-Angle
Right-AngleTrigonometry
Trigonometry
• How do we understand and use
trigonometric relationships of acute
angles in triangles?
• How do we determine side lengths of
right triangles by using trigonometric
functions?
HoltMcDougal
Algebra 2Algebra 2
Holt
Right-Angle Trigonometry
The reciprocals of the sine, cosine, and tangent
ratios are also trigonometric ratios. They are
trigonometric functions, cosecant, secant, and
cotangent.
Holt McDougal Algebra 2
Right-Angle Trigonometry
Finding All Trigonometric Functions
1. Find the values of the six trigonometric functions for θ.
Find the length of the hypotenuse.
a2 + b2 = c2
c2 = 242 + 702
c2 = 5476
c = 74
Pythagorean Theorem.
Substitute 24 for a and 70 for b.
Simplify.
Solve for c. Eliminate
the negative solution.
hyp.
74
70
opp.
θ
Find the lengths of the 6 trigonometric values.
adj. 24
opp 70 35
24 12
adj
opp 70 35
sin  


cos  
tan  




hyp
hyp
adj
74 37
74 37
24 12
1
37
csc  

sin 
35
Holt McDougal Algebra 2
37
1

sec  
cos 
12
cot  
12
1

tan  35
Right-Angle Trigonometry
Helpful Hint
In each reciprocal pair of trigonometric functions,
there is exactly one “co”
Holt McDougal Algebra 2
Right-Angle Trigonometry
Finding All Trigonometric Functions
2. Find the values of the six trigonometric functions for θ.
Find the length of the hypotenuse.
a2 + b2 = c2
c2 = 182 + 802
c2 =6724
c = 82
Pythagorean Theorem.
Substitute 18 for a and 80 for b.
Simplify.
Solve for c. Eliminate
the negative solution.
hyp.
82
80
opp.
θ
Find the lengths of the 6 trigonometric values.
adj. 18
opp 80 40
9
adj 18
opp 80 40
sin  


cos  
tan  




hyp 82
hyp 82
41
adj 18
41
9
1
41
csc  

sin 
40
Holt McDougal Algebra 2
41
1

sec  
cos 
9
cot  
9
1

tan  40
Right-Angle Trigonometry
Sports Application
3. In a waterskiing competition,
a jump ramp has the measurements
shown. To the nearest foot, what
opp.
is the height h above water that a
skier leaves the ramp?
hyp.
Substitute 15.1° for θ, h for
opp., and 19 for hyp.
Multiply both sides by 19.
5≈h
Use a calculator to simplify.
The height above the water is about 5 ft.
Holt McDougal Algebra 2
Right-Angle Trigonometry
Sports Application
4. A skateboard ramp will have a height
of 12 in., and the angle between the
ramp and the ground will be 17°.
To the nearest inch, what will be
the length l of the ramp?
hyp.
opp.
Substitute 17° for θ, l for hyp.,
and 12 for opp.
Divide 12 by sine 17.
l ≈ 41
Use a calculator to simplify.
The length of the ramp is about 41 in.
Holt McDougal Algebra 2
Right-Angle Trigonometry
When an object is above or below another object,
you can find distances indirectly by using the angle
of elevation or the angle of depression between the
objects.
Holt McDougal Algebra 2
Right-Angle Trigonometry
Geology Application
5. A biologist whose eye level is 6 ft above the
ground measures the angle of elevation to the
top of a tree to be 38.7°. If the biologist is
standing 180 ft from the tree’s base, what is opp.
the height of the tree to the nearest foot?
Which function
relates the opposite
and the adjacent?
180(tan 38.7°) = x
144 ≈ x
h ≈ 150
Substitute 38.7 for θ, x for opp.,
and 180 for adj.
Multiply both sides by 180.
Use a calculator to simplify.
Add 6 for the biologist’s height.
The height of the tree is about 150 ft.
Holt McDougal Algebra 2
adj.
Right-Angle Trigonometry
Geology Application
6. A surveyor whose eye level is 6 ft above the ground
measures the angle of elevation to the top of the
x
highest hill on a roller coaster to be 60.7°. If the opp.
surveyor is standing 120 ft from the hill’s base,
what is the height of the hill to the nearest foot?
Which function
relates the opposite
and the adjacent?
120(tan 60.7°) = x
60.7°
120 ft adj.
Substitute 60.7 for θ, x for opp.,
and 120 for adj.
Multiply both sides by 120.
214 ≈ x Use a calculator to simplify.
h ≈ 220 Add 6 for the surveyor’s height.
The height of the hill is about 220 ft.
Holt McDougal Algebra 2
Right-Angle Trigonometry
Lesson 10.1 Practice B
Holt McDougal Algebra 2