Download find sine and tangent. - Fort Thomas Independent Schools

P.o.D. – Consider triangle ABC
with C=90 degrees. (on the
board)
1.) If A=28 degrees and b=7, find
c.
2.) If c=17 and a=15, find b.
3.) If B=45 degrees and c=23, find
a.
4.) If A=76 degrees and a=1, find
b.
5.) If a=6 and b=8, find c.
1.) 7.93
2.) 8
3.) 16.26
4.) 0.25
5.) 10
10-2: More Right Triangle
Trigonometry
Learning Target: I can determine
the measure of an angle given its
sine, cosine, or tangent; solve
real-world problems using right
triangle trig.
EX: Consider a right triangle in
which the hypotenuse is 19 and
one leg is 15. What is the
measure of the angle formed
between these two sides?
(draw a picture on the
whiteboard)
15
cos 𝜃 =
→
19
15
θ = cos
19
≈ 37.86°
−1
EX: Find the measures of all 3
angles in a triangle with side 3,
4, and 5.
(draw a picture on the
whiteboard)
4
tan 𝜃 = →
3
4
−1
𝜃 = tan
3
= 53.13°
3
tan 𝛼 = →
4
3
−1
𝛼 = tan
4
= 36.87°
𝛽 = 180 − 53.13 − 36.87 = 90°
EX: Find all 3 angles for the
triangle below: b=18, c=33
B
c=33
A
b=18
C
18
cos 𝐴 =
→
33
18
−1
𝐴 = cos
33
≈ 56.94°
18
sin 𝐵 =
→
33
18
−1
𝐵 = sin
≈ 33.06°
33
𝐶 = 180 − 56.94 − 33.06 = 90°
*Notice that angles A and B will
always be complementary:
56.94+33.06=90.
3
EX: If cos 𝜃 = , find sine and
4
tangent.
*begin by drawing a triangle.
A
4
𝜃
C
B
3
*First, find the missing leg.
32 + 𝑏 2 = 42 → 𝑏 2 = 16 − 9 = 7 →
𝑏 = √7
√7
sin 𝜃 =
4
√7
tan 𝜃 =
3
Let’s review the properties of a
30-60-90 triangle.
A
60°
2x
x
30°
B
C
𝑥√3
Leg opposite 30 = x
Leg opposite 60 = 𝑥 √3
Hypotenuse = 2x
EX: Find the sine, cosine, and
tangent of 30 degrees.
𝑥
1
sin 30° =
=
2𝑥 2
𝑥 √3 √3
cos 30° =
=
2𝑥
2
𝑥
tan 30° =
=
𝑥 √3
1
=
√3
1
∙
√3
√3 √3
=
√3
3
EX: Find the sine, cosine, and
tangent of 60 degrees.
𝑥 √3 √3
sin 60° =
=
2𝑥
2
𝑥
1
cos 60° =
=
2𝑥 2
𝑥 √3
tan 60° =
= √3
𝑥
*Notice that sin 30° = cos 60° and
sin 60° = cos 30°.
These are known as cofunctions, because they are
complements of one another.
Let’s review the properties of a
45-45-90 triangle.
45
𝑥√2
x
45
x
EX: Find the sine, cosine, and
tangent of 45 degrees.
𝑥
1
√2
sin 45° =
=
=
2
𝑥 √2 √2
𝑥
1
√2
cos 45° =
=
=
2
𝑥 √2 √2
𝑥
tan 45° = = 1
𝑥
EX: A 115-ft ladder attached to a
fire truck rests against the side
of a building so that the top of
the ladder is 111-ft above the
ground. Find the angle formed
by the truck and the ladder.
115
111
𝜃
111
sin 𝜃 =
→
115
111
−1
𝜃 = sin
≈
115
74.84°
Angle of Depression vs. Angle of
Elevation
(draw a picture on the
whiteboard)
EX: A hawk, cruising at an
altitude of 215 feet, spots a prey
at a direct distance of 406 feet.
Find the angle of depression of
the prey as seen by the hawk.
hawk
𝜃
406
215
𝜃
215
sin 𝜃 =
→
406
215
−1
𝜃 = sin
≈
406
31.98°
*Remember, angle of depression
is congruent to angle of
elevation.
Upon completion of this lesson,
you should be able to:
1. Solve right triangles using
trig functions.
2. Identify the properties of a
45/45/90 and 30/60/90
triangle.
3. Solve story problems
involving right triangles.
For more information, visit
http://www.themathpage.com/atrig/solveright-triangles.htm
HW Pg.673 2-9, 11-19