Math 151
Appendix D
Trigonometry Review
Measurement of Angles Angles can be measured in degrees or radians. One complete revolution
corresponds to 360°, or 2π radians. So, 360° = 2π radians which gives the conversion formula:
π radians = 180°
Example:
A.
Convert 72° to radians.
B.
Convert !
3!
to degrees.
4
Trigonometric Ratios The hypotenuse of a right triangle is the longest side and is opposite the
right angle. Trigonometric functions are defined as ratios of the lengths of the sides of a right
triangle. These are the basis for the trigonometric functions in which we replace the triangle by
points on a plane.
a length of opposite side opp
sin( A) = =
=
c
length of hypotenuse
hyp
b length of adjacent side adj
=
=
c
length of hypotenuse
hyp
a length of opposite side opp
tan( A) = =
=
b length of adjacent side
adj
cos( A) =
c
length of hypotenuse
hyp
=
=
a length of opposite side opp
c
length of hypotenuse
hyp
sec( A) = =
=
b length of adjacent side adj
b length of adjacent side
adj
cot( A) = =
=
a length of opposite side opp
csc( A) =
Pythagorean Theorem For any right triangle, with hypotenuse c and sides a and b,
c2 = a2 + b2.
Math 151
30-60-90 Triangle In a 30°-60°-90° triangle, the length of the side opposite the 30° angle is half
the length of the hypotenuse.
1
2
1
60°
2
1
60°
1
1
60° 60°
1
2 2
30°30°
30° 30°
3
2
3
2
3
45-45-90 (Isosceles Right) Triangle
2
2
45°
45°
1
2
1
2 45°
2
45°
1
The Unit Circle
3
2
3
Math 151
Example: Find all trig ratios for ! =
Example: If cos x =
2"
.
3
1
!
and 0 ! x ! , find the values of the other trig functions evaluated at x.
5
2
Trigonometric Identities In addition to the values on the Unit Circle, you will be expected to
readily recall the following identities:
The Pythagorean identities:
sin 2 ! + cos 2 ! = 1
1+ tan 2 ! = sec 2 !
1+ cot 2 ! = sin 2 !
The quotient and reciprocal identities:
sec! =
1
cos!
csc! =
1
sin !
tan ! =
sin !
cos!
The double-angle formulas:
sin (2!) = 2sin ! cos!
cos(2!) = 2cos 2 ! !1
"
cos(2!) = 1! 2sin 2 !
"
1
(1+ cos(2!))
2
1
sin 2 ! = (1! cos(2!))
2
cos 2 ! =
Example: Solve the following equations for x, where 0 ! x ! 2! .
A.
2cos x !1 = 0
B.
2cos x + sin 2x = 0
cot ! =
cos!
1
=
sin ! tan !
Math 151
Example: If sec x =
5
!
and ! < x < 0 , what is the value of sin 2x ?
3
2
Graphs of Sine, Cosine, and Tangent Functions
Example: Sketch a graph of f ( x) = 1! cos x .
"
!%
Example: Sketch a graph of f ( x) = tan $$ x ! ''' .
$#
2 '&