Embed Size (px)
DESCRIPTION
Section 4.3. Right Triangle Trigonometry. Overview. In this section we apply the definitions of the six trigonometric functions to right triangles. Before we do that, however, let’s remind ourselves about the Pythagorean Theorem:. A Picture. Example. - PowerPoint PPT Presentation
Citation preview
Section 4.3
Right Triangle Trigonometry
Overview
• In this section we apply the definitions of the six trigonometric functions to right triangles.
• Before we do that, however, let’s remind ourselves about the Pythagorean Theorem:
A Picture
Example
• Find the missing side of the right triangle.
?
4.3 – Right Triangle Trigonometry
SOHCAHTOA
“Some Old Hippie Came Around Here Trippin’ On Acid.”“Some Old Hog Came Around Here and Took Our Apples
adj.
opp.θtan
hyp.
opp.θsin
hyp.
adj.θ cos
SOH-CAH-TOAThe six trigonometric functions
of the acute angle are…
adj
opp
hyp
adj
hyp
opp
tan
cos
sin
opp
adj
adj
hyp
opp
hyp
cot
sec
csc
4.3 – Right Triangle TrigonometryFind the six trig functions of θ in the triangle
below.
12
13
12
5θtan
5
12θcot
12
13θsec
5
13θ csc
13
5θsin
13
12θ cos
5
An Example
State the six trigonometric values for angles C and T.
4.3 – Right Triangle TrigonometryFind the sine, cosine, and tangent of 45º using the triangle below.
1
45º 1
11
154tan
2
2
2
154sin
2
2
2
145 cos
2
Construct a 45-45-90 triangle with hypotenuse=1. Find the sine, cosine, and
tangent of 45º using your triangle.
1
Finding sides for a 30 – 60 – 90 TriangleGiven: Equilaterial triangle of side length 1, and altitude h.
We know form geometry that the altitude h, bisects the angle it is drawn from and that it is the perpendicular bisector of the opposite side.
1
11h
30⁰
1
2
1
2
60⁰
What is the length of the altitude h?30⁰
60⁰
Find the sine, cosine, and tangent of 30º and 60º using the triangle from the last slide.
30º
60º
2
360sin
2
160 cos
31
360tan
2
130sin
2
330 cos
3
3
3
130tan
Summary
Deg. Rad. Sin Cos Tan
0º
30º
45º 1
60º
90º
180º
270º
360º
21
22
23
21
22
23
33
3
Cofunctions
• The sine of an angle is equal to the cosine of its compliment (and vice versa).
• The tangent of an angle is equal to the cotangent of its compliment (and vice versa).
• The secant of an angle is equal to the cosecant of its compliment (and vice versa).
4.3 – Right Triangle Trigonometry
Cofunctions sin 30º =
cos 60º =
sin 30º = cos (90º – 30º)
tan 57º 1.5399
cot 33º ?
cot 33º = tan(90º – 33º) = tan 57º 1.5399
21
21
Solving Right Triangles
1. Write the appropriate trigonometric relationship for the unknown value (there may be more than one).
2. Use your scientific calculator to find the appropriate trigonometric value or angle (make sure your calculator is in degree mode).
Examples
4.3 – Right Triangle Trigonometry
The angle of elevation from point X to point Y (above X) is made between the ray XY and a horizontal ray.
The angle of depression from point X to point Z (below X) is made between the ray XZ and a horizontal ray.
Y
a.o.e. X a.o.d.
Z
4.3 – Right Triangle Trigonometry
At a certain time of day Giant Sam’s shadow is 400 feet long. If the angle of elevation of the sun is 42º, how tall is he?
42º
400 ft.
h
400
h42tan
h = 400 tan 42º
h = 400(.9004)
h = 360.2 ft.
