Brief Review of Trigonometry
1. (a) An angle is determined by rotating a ray about its endpoint.( counterclockwise rotation measures a positive
angle, clockwise rotation measures a negative angle.) Angles are measured by either degrees or radians. In Calculus,
radian measure is used. One degree(1°) is the measure of a positive angle formed by 1/360 of one complete
revolution and one radian is the measure of a positive angle which intercepts an arc S which is equal in length
to the radius r of the circle. Hence, a central angle of θ radians intercepts an arc of length θ times the radius.
Formula for arc length S = r θ where θ is measured in radians.
(b) The area of a circular sector is a fractional part (θ
θ/2π
π) of the area of the circle (π
πr2 )
Hence A =( θ/2π
π ) (π
π r2 ) = r2θ/ 2 for the area of a circular sector.
(c) An angle of 1° subtends an arc which is 1/360 of the circumference ( 2π
π) of the unit circle so that
1° = ( 1/360) (2π
π) radians = (π
π,180) radians
and
1 radian = (180/π
π) degrees.
Example 30° = 30 ( π/180 radians) = π/6 radians.
2. You must be able to find in exact form( no calculators)the trig functions for the special
and quadrantal angles measured in radians. Below are some guidelines for remembering these.
(a) The circular functions are defined on a unit circle as follows:
sin θ
y
sin θ = y
cos θ = x
tan θ =
=
y
x
cos θ
θ
x
1
1
1
1
1
x
x
csc θ =
=
sec θ =
=
cot θ =
=
y
x
y
sin θ
cos θ
tan θ
By similar triangles we have the following definitions on a circle of radius r
and can relate this to the right triangle involved:
opp
x
adj
y
opp
y
=
cos θ =
=
tan θ =
=
sin θ =
r
hyp
r
hyp
x
adj
r
hyp
r
hyp
x
adj
csc θ =
=
sec θ =
=
cot θ =
=
y
opp
x
adj
y
opp
(b) The triangles below from geometry can be used to remember the
values of the trig functions at 30°(π
π/6 rad), 45° ( π/4 rad), 60° ( π/3 rad)
Example : sin 30° = sin (π
π/6 radians) = opp/hyp = 1/2
tan 45° = tan ( π/4 radians) = opp/adj = 1/1 = 1
π/6
cos 60° = cos(π
π/3 radians) = adj/hyp = 1/2
r
θ
y
x
2
2
1
1
π/4
1
1
3
(c) For angles which are multiples of π/6 , π/4, π/3 we use the concept of reference angles. The reference angle
( called α here ) is the acute angle formed by the terminal side of the given angle and the x-axis. The value for a
circular function of the given angle θ is the same in absolute value as the value of the circular function of the
reference angle. The sign of the circular function of the given angle is determined by the particular quadrant
in which the terminal side of the given angle lies. Since α will be an acute angle, we can use the triangles above.
Quadrant II Here α is the reference angle and α = π – θ
sin θ = ± sin α = + sin α
sin 2π
π/3 = + sin π/3 = + 3 /2
cos θ = ± cos α = – cos α
cos 5π
π/6 = – cos π/6 = – 3 /2
tan θ = ± tan α = – tan α
tan 3π
π/4 = – tan π/4 = – 1/1 = 1
Quadrant III Here α is the reference angle α = θ – π
sin θ = ± sin α = – sin α
sin 7π
π/6 = – sin π/6 = – 1 / 2
cos θ = ± cos α = – cos α
cos 5π
π/4 = – cos π/4 = – 2 / 2
tan θ = ± tan α = + tan α
tan 4π
π/3 = +tan π/3 = + 3 / 1 =
Quadrant IV Here α is the reference angle α = 2π
π–θ
sin θ = ± sin α = – sin α
sin 5 π/3 = – sin π/3 = – 3 / 2
cos θ = ± cos α = + cos α cos 11π
π/ 6 = + cos π/6 = + 3 / 2
tan θ = ± tan α = – tan α tan 7π
π/4 = – tan π/4 = – 1 / 1 = –1
3. For the quandrantal angles: 0, π/2, π, 3π
π/2, 2π
π, etc. the graphs are useful.
4. The trigonometric functions evaluated at the real number t are defined to have the same values as the corresponding
circular functions for an angle of t radians.
Ex: sin 2 =sin(2 rad) =+ sin(π
π–2)rad = + sin(π
π – 2). Here t is 2 and the reference angle α is π–2 since π/2 ≤ 2 ≤ π
5. (a) An even function has the property that f (–x) = f ( x ). The cosine function is even so cos (–x) = cos ( x ).
π
π
Ex. cos (–
) = cos ( ) = 3 / 2
6
6
(b) An odd function has the property that f (– x ) = – f ( x ). The sine function is odd so sin ( – x) = – sin ( x)
π
π
Ex. sin( – ) = – sin ( ) = – 3 / 2
3
3
6. With the guidelines above you should be able to evaluate the trig functions for the chart below without a calculator.
θ
0 π /6 π /4 π /3 π /2 2π
π /3 3π
π /4 5π
π /6 π 7π
π /6 5π
π /4 4π
π /3 3π
π /2 5π
π /3 7π
π /4 11π
π /6
2π
π
sin θ
cos θ
tan θ
7. Trig identities that you must know:
Pythagorean identities
Double Angle identities
2
2
sin θ + cos θ = 1
sin 2 θ = 2 sin θ cos θ
2
2
tan θ + 1 = sec θ
cos 2 θ = cos2 θ – sin2 θ
cot2 θ + 1 = csc2 θ
1 + cos 2θ
θ
cos2 θ =
2
1 – cos 2θ
θ
sin2 θ =
2
8. Identities used less often
Sums and differences
sin ( θ1 ± θ2 ) = sin θ1 cos θ2 ± sin θ2 cos θ1
cos (θ
θ1 ± θ2 ) = cos θ1 cos θ2
tan (θ
θ1 ± θ2 ) =
∓ sin θ1 sin θ2
tan θ1 ± tan θ2
1 ∓ tan θ1tan θ2