Trigonometric Functions of Any Angle
Definitions of Trigonometric Functions of Any Angle
Let
be an angle in standard position with
a point on the terminal side of
and
.
Since
cannot be zero, the sine and cosine functions are defined for any real value of
. However, when
, the tangent and secant of
are undefined (e.g.,
is undefined). Similarly, when
, the cotangent and cosecant of
are undefined.
The signs of the trigonometric functions in the four quadrants follow from their definitions. For instance,
is positive wherever
, which occurs in Quadrants I and IV. (Remember,
is always positive.)
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'
Trigonometric Functions of Quadrantal Angles
Quadrantal angles are angles whose terminal sides lie along the
-axis or
-axis. These angles include
,
,
, and
.
The values of the sine, cosine, and tangent functions at these angles are as follows:
0 (pi)/(2) pi (3pi)/(2)
sin theta {:[sin 0=(y)/(r)],[=(0)/(1)=0]:} {:[sin (pi)/(2)=(y)/(r)],[=(1)/(1)=1]:} {:[sin pi=(y)/(r)],[=(0)/(1)=0]:} {:[sin (3pi)/(2)=(y)/(r)],[=(-1)/(1)=-1]:}
cos theta {:[cos 0=(x)/(r)],[=(1)/(1)=1]:} {:[cos (pi)/(2)=(x)/(r)],[=(0)/(1)=0]:} {:[cos pi=(x)/(r)],[=(-1)/(1)=-1]:} {:[cos (3pi)/(2)=(x)/(r)],[=(0)/(1)=0]:}
tan theta {:[tan 0=(y)/(x)],[=(0)/(1)=0]:} {:[tan (pi)/(2)=(y)/(x)],[=(1)/(0)]:}<br> => undefined {:[tan pi=(y)/(x)],[=(0)/(-1)=0]:} {:[tan (3pi)/(2)=(y)/(x)],[=(-1)/(0)]:}<br> => undefined
The tangent function is undefined at
and
because at these angles,
, making the denominator of
equal to zero.
Reference Angles
The values of trigonometric functions for angles greater than
(or less than
) can be determined using corresponding acute angles, called reference angles.
Let
be an angle in standard position. Its reference angle
is the acute angle formed by the terminal side of
and the horizontal axis.
(radians)
(degrees)
(radians)
(degrees)
(radians)
(degrees)
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' <figcaption> theta^(')=2pi-theta (radians)<br> theta^(')=360^(@)-theta (degrees) </figcaption>
Trigonometric Values of Common Angles
For a right triangle with acute angle
and sides of lengths
and
,
Thus,
and
are equal except possibly in sign. The same holds for
and
, as well as the other four trigonometric functions. The quadrant in which
lies determines the sign of the function value.
Evaluating Trigonometric Functions of Any Angle
To evaluate a trigonometric function of any angle
:
Determine the function value of the associated reference angle
.
Assign the appropriate sign based on the quadrant in which
lies.
Using reference angles and known special angles significantly expands the range of exact trigonometric function values. For example, knowing the function values of
allows one to determine function values for all angles with
as a reference angle.
Exact Trigonometric Values
(degrees)
(radians)
undefined
undefined
theta (degrees) 0^(@) 30^(@) 45^(@) 60^(@) 90^(@) 180^(@) 270^(@)
theta (radians) 0 (pi)/(6) (pi)/(4) (pi)/(3) (pi)/(2) pi (3pi)/(2)
sin theta 0 (1)/(2) (sqrt2)/(2) (sqrt3)/(2) 1 0 -1
cos theta 1 (sqrt3)/(2) (sqrt2)/(2) (1)/(2) 0 -1 0
tan theta 0 (sqrt3)/(3) 1 sqrt3 undefined 0 undefined
Memorization Pattern for
and
This pattern provides an easy way to remember the values for sine and cosine at common angles.
