Download Trigonometric Functions t t. t t. t t. t t. t t. t

Trigonometric Functions
The sine function is f(t) = sin(t) or f(t) = sint.
The cosine function is f(t) = cos(t) or f(t) = cost.
The tangent function is f(t) = tan(t) or f(t) = tant.
The cosecant function is f(t) = csc(t) or f(t) = csct.
The secant function is f(t) = sec(t) or f(t) = sect.
The cotangent function is f(t) = cot(t) or f(t) = cott.
t is an angle in standard position in radians or degrees.
sin(t) or sint is not sin times t, it is the sine function evaluated at t.
General Background
There are two interconnected ways to define the trigonometric functions.
1. Using the unit circle (or any circle).
Angles measured in radians.
2. Using right triangles.
Angles measured in degrees, so must convert to radians.
The Unit Circle
The unit circle is the circle of radius 1 with its center at the origin.
The unit circle has the equation x2 + y2 = 1.
The terminal side of every central angle, t, in standard position intersects the unit circle at the point (x,
y).
The trigonometric functions are defined as follows:
sint = y
cost = x
tant = y/x provided x ≠ 0
csct = 1/y provided y ≠ 0
sect = 1/x provided x ≠ 0
cott = x/y provided y ≠ 0
An Arbitrary Circle
A circle of radius r centered at the origin has the equation x2 + y2 = r2.
The terminal side of every central angle, t, in standard position intersects the circle at the point (x, y).
The trigonometric functions are defined as follows:
sint = y/r
cost = x/r
tant = y/x provided x ≠ 0
csct = r/y provided y ≠ 0
sect = r/x provided x ≠ 0
cott = x/y provided y ≠ 0
The Right Triangle
Let t be an acute angle in a right triangle with the sides defined as shown.
hypotenuse
opposite
t
adjacent
The trigonometric functions are defined as follows:
sin t
opposite
hypotenuse
cos t
adjacent
hypotenuse
tan t
opposite
adjacent
csc t
hypotenuse
opposite
sec t
hypotenuse
adjacent
cot t
adjacent
opposite
When this triangle is put in standard position,
1. The hypotenuse is the terminal side and is 1 for the unit circle or r for an arbitrary circle.
2. The adjacent side is x.
3. The opposite side is y.
45°-45°-90° Triangle
The hypotenuse can be any length, but if we use the unit circle, then it will be 1.
Since this is an isosceles triangle, the opposite side and the adjacent side are equal.
2
Using the Pythagorean Theorem, the opposite and adjacent sides are
.
2
For the angle t = 45o = π/4, x = y =
2
.
2
30°-60°-90° Triangle
The hypotenuse can be any length, but if we use the unit circle, then it will be 1.
Since this is half an equilateral triangle, the hypotenuse is twice the short side.
The short side is ½.
The short side is the opposite side when t = 30o.
The short side is the adjacent side when t = 60o.
3
Using the Pythagorean Theorem, the last side is
.
2
For the angle t = 30o = π/6, x =
3
and y = ½.
2
For the angle t = 60o = π/3, x = ½ and y =
3
.
2
Reference Angle
A reference angle is the acute angle formed by the terminal side of an angle and the x-axis.
1. If t is in the second quadrant, then the reference angle is π – t.
2. If t is in the third quadrant, then the reference angle is t – π.
3. If t is in the fourth quadrant, then the reference angle is 2π – t.
You only need to know the angles between 0 and π/2.
You need to know the sign of x and y in each quadrant.
The angles of 3π/4, 5π/4, and 7π/4 have a reference angle of π/4.
2
2
For the angle t = 3π/4, x = –
and y =
.
2
2
2
For the angle t = 5 π/4, x = y = –
.
2
2
2
For the angle t = 7 π/4, x =
and y = –
.
2
2
The angles of 5π/6, 7π/6, and 11π/6 have a reference angle of π/6.
3
For the angle t = 5π/6, x = –
and y = ½.
2
3
For the angle t = 7π/6, x = –
and y = –½.
2
3
For the angle t = 11π/6, x =
and y = –½.
2
The angles of 2π/3, 4π/3, and 5π/3 have a reference angle of π/3.
3
For the angle t = 2π/3, x = –½ and y =
.
2
3
For the angle t = 4π/3, x = –½ and y = –
.
2
3
For the angle t = 5π/3, x = ½ and y = –
.
2
Trigonometric Functions Defined for Specific Angles
You need to know these values.
If you know the reference angles, then you only need to know 0 to π/2.
If you know the definitions, then you only need to know sint and cost.
t
0
sint
0
cost
tant
6
1
2
4
3
2
2
1
3
2
2
2
3
2
1
2
0
3
3
1
3
2
2
2 3
3
2 3
3
2
3
1
csct und
sect
1
cott und
2
1
2
3
3
4
3
2
1
2
2
2
5
6
1
2
7
6
1
2
0
2
2
-
3
2
-1
und - 3
-1
-
3
3
0
3
3
1
2 3
3
2
2
und
-2
2
und
-2
- 2 -
2 3
3
3
3
0
0
-
3
3
-
-1
- 3
-
-1 und
3
2
-
2
2
-
2
2
3
2
1
2
1
-
- 2 -
2 3
- 2
3
3
4
3
5
4
1
5
3
3
2
-1
-
3
2
2
2
11
6
1
2
1
2
1
3
3
0
-2
und
7
4
-
0
3
2
2
2
3
und
- 3
-1
2 3
3
-1
2 3
3
- 2
-2
und
3
3
0
-
-
-
2
0
2
2
2 3
3
1
3
3
-1
- 3
und
Examples
Find the six trig. functions for t = 5π/2, 17π/6, 13π/4, and 14π/3.
Find the six trig. functions for t = -3π/2, -9π/4, -13π/3, and -7π/6.
Find the six trig. functions when (-3, 4) is on the terminal side of angle t.
Find the six trig. functions when (5, 12) is on the terminal side of angle t.
Find the six trig. functions when (-4, 7) is on the terminal side of angle t.
Domain and Range
Domain
f(x) = sinx
ℝ
f(x) = cosx
ℝ
n
where n is an odd integer
x
f(x) = tanx
2
f(x) = cscx
x n where n is an integer
n
where n is an odd integer
x
f(x) = secx
2
f(x) = cotx
x n where n is an integer
Range
[-1, 1]
[-1, 1]
ℝ
(-∞, -1]∪[1, ∞)
(-∞, -1]∪[1, ∞)
ℝ
Period
A function is periodic if f(x + c) = f(x) for all x. If c is the smallest value that f is periodic, then c is the
period.
Sine, cosine, secant, and cosecant have a period of 2π.
Tangent and cotangent have a period of π.
Identity
An identity is an equation that is always true.
a. Reciprocal Identities
1
csc x
sin x
b. Quotient Identities
sin x
tan x
cos x
c. Pythagorean Identities
sin2x + cos2x = 1
sin2x = (sinx)2
sinx2 = sin(x2)
d. Co-function Identities
sin(π/2 – x) = cosx
csc(π/2 – x) = secx
sec x
1
cos x
cot x
cos x
sin x
cot x
1
tan x
1 + cot2x = csc2x
tan2x + 1 = sec2x
cos(π/2 – x) = sinx
sec(π/2 – x) = cscx
tan(π/2 – x) = cotx
cot(π/2 – x) = tanx
Calculators
1. Make sure the calculator is in the right mode.
DEG represents degree mode.
RAD represents radian mode.
2. For cosecant, secant, and cotangent, use the reciprocal identities.
Applications
A man is 135 ft from the base of a broadcasting tower. The angle of elevation is 63.4o. How tall is the
tower?
A biologist needs to know the width of a river for an experiment. From a point A on his side opposite a
point B on the other, he walks 100 ft to point C. He determines angle ACB has measure of 54°. How
wide is the river?