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Lesson 43:
Sine, Cosine, and Tangent,
Inverse Functions
Here we show three triangles. Each one
is a right triangle, and each one contains
a 30° angle.
4
2
30°
120
30°
3.464
103.92
60
240
30°
207.85
120
The longest side of a right triangle
is always the side opposite the right
angle, and this side is called the
hypotenuse. The hypotenuses of
these triangles from left to right are
4, 120, and 240 units long.
We remember that right triangles that
also have one equal acute angle are
similar triangles and that the ratios of
corresponding sides of similar triangles
are equal. If for each of the three triangles
shown we write the ratio of the side
opposite the 30° angle to the hypotenuse,
we get these expressions, each of which
has a value of 0.5.
2/4
60/120
120/240
We would find this to be true for this
ratio in any right triangle with a 30°
angle. In a right triangle we call this
ratio, which is the ratio of the side
opposite the angle to the hypotenuse,
the sine of the angle.
sin 30° = side opposite 30° angle
hypotenuse
This ratio always equals 0.5 for a
30° angle. Every angle has a specific
value for the ratio of the opposite
side to the hypotenuse.
In a right triangle the ratio of the side
adjacent to the angle to the hypotenuse
is called the cosine of the angle.
cos 30° = side adjacent 30° angle
hypotenuse
We call the ratio of the side
opposite the angle to the side
adjacent to the angle the tangent of
the angle.
tan 30° = side opposite 30°
side adjacent 30°
SOH
CAH
TOA
Sin = Opp.
Cos = Adj.
Tan = Opp.
Hyp.
Hyp.
Adj.
The sines, cosines, and tangents of
angles can be obtained by using the
“sin”, “cos”, and “tan” keys on a
scientific calculator. To find the
proper values when the angle is
measured in degrees, the calculator
must be in degree mode.
Calculators will give an estimate of the
value of a trigonometric function to
more than five decimal places. To find
the sine of 39.2°, firs put the
calculator in degree mode (press the
“deg” or “drg” key). Then enter
“39.2”, then “sin”.
sin 39.2° ≈ .63
Example:
Find the cosine and tangent of
39.2°.
Answer:
Cos 39.2° ≈ .775
Tan 39.2° ≈ .816
Example:
Solve 4 cos 57.2°
Answer:
2.17
Example:
Find the sine of A, the cosine of B and
the tangent of C. Round answers to two
decimal places.
7
4
8.6
A
5.74
5
6
10
B
C
7
8
Answer:
Sin A = 4/7 ≈ 0.57
Cos B = 7/8.6 ≈ 0.81
Tan C = 6/8 ≈ 0.75
The inverse sine of a number is the angle
whose sine is the number. Several
examples will help.
a)
The sine of 30° is 0.5, so the inverse
sine of 0.5 is 30°
b)
The cosine of 30° is approximately
0.866, so the inverse cosine of 0.866
is 30°
c)
The tangent of 30° is approximately
0.577, so the inverse tangent of 0.577
is 30°
To find the inverse sine, cosine and
tangent of a number, we use the
“inv” key followed by the
trigonometric function key.
Example:
Find (a) the angle whose sine is
0.643 and (b) the angle whose
cosine is 0.216.
Answer:
a) ≈ 40°
b) ≈ 77.53°
HW: Lesson 43 #1-30