Introduction
The six trigonometric functions (sine, cosine, tangent,
cosecant, secant, and cotangent) can be used to find the
length of the sides of a triangle or the measure of an
angle if the length of two sides is given. Previously these
functions could only be applied to angles up to 90°.
However, by using radians and the unit circle, these
functions can be applied to any angle.
1
5.1.4: Evaluating Trigonometric Functions
Key Concepts
• Recall that sine is the ratio of the length of the
opposite side to the length of the hypotenuse, cosine
is the ratio of the length of the adjacent side to the
length of the hypotenuse, and tangent is the ratio of
the length of the opposite side to the length of the
adjacent side. (You may have used the mnemonic
device SOHCAHTOA to help remember these
relationships: Sine equals the Opposite side over the
Hypotenuse, Cosine equals the Adjacent side over the
Hypotenuse, and Tangent equals the Opposite side
over the Adjacent side.)
2
5.1.4: Evaluating Trigonometric Functions
Key Concepts, continued
• Three other trigonometric functions, cosecant, secant,
and cotangent, are reciprocal functions of the first
three. Cosecant is the reciprocal of the sine function,
secant is the reciprocal of the cosine function, and
cotangent is the reciprocal of the tangent function.
3
5.1.4: Evaluating Trigonometric Functions
Key Concepts, continued
• The cosecant of  = csc  =
length of hypotenuse
1
; csc q =
length of opposite side
sinq
• The secant of  = sec  =
length of hypotenuse
1
; sec q =
length of adjacent side
cosq
• The cotangent of  = cot  =
length of adjacent side
1
; cot q =
length of opposite side
tanq
5.1.4: Evaluating Trigonometric Functions
4
Key Concepts, continued
•
The quadrant in which the terminal side is
located determines the sign of the trigonometric
functions. In Quadrant I, all the trigonometric
functions are positive. In Quadrant II, the sine
and its reciprocal, the cosecant, are positive and
all the other functions are negative. In Quadrant
III, the tangent and its reciprocal, the cotangent,
are positive, and all other functions are negative.
In Quadrant IV, the cosine and its reciprocal, the
secant, are positive, and all other functions are
negative.
5.1.4: Evaluating Trigonometric Functions
5
Key Concepts, continued
•
You can use a mnemonic device to remember in
which quadrants the functions are positive: All
Students Take Calculus (ASTC).
6
5.1.4: Evaluating Trigonometric Functions
Key Concepts, continued
• However, instead of memorizing this, you can also think
it through each time, considering whether the opposite
and adjacent sides of the reference angle are positive
or negative in each quadrant.
• To find a trigonometric function of an angle given a point
on its terminal side, first visualize a triangle using the
reference angle. The x-coordinate becomes the length of
the adjacent side and the y-coordinate becomes the
length of the opposite side. The length of the hypotenuse
can be found using the Pythagorean Theorem.
Determine the sign by remembering the ASTC pattern or
by considering the signs of the x- and y-coordinates.
5.1.4: Evaluating Trigonometric Functions
7
Key Concepts, continued
• To find the trigonometric functions of special angles,
first find the reference angle and then use the pattern
to determine the ratio.
• For angles larger than 2 radians (360°), subtract
2 radians (360°) to find a coterminal angle, an
angle that shares the same terminal side, that is less
than 2 radians (360°). Repeat if necessary.
• For negative angles, find the reference angle and then
apply the same method.
8
5.1.4: Evaluating Trigonometric Functions
Common Errors/Misconceptions
• using the incorrect trigonometric ratio
• forgetting to consider whether the trigonometric ratios
are negative
• mistaking the quadrants in which each trigonometric
function is positive
9
5.1.4: Evaluating Trigonometric Functions
Guided Practice
Example 2
Find sin  if  is a positive angle in standard position
with a terminal side that passes through the point (5, –2).
Give an exact answer.
10
5.1.4: Evaluating Trigonometric Functions
Guided Practice: Example 2, continued
1. Sketch the angle and draw in the triangle
associated with the reference angle.
Recall that a positive angle is created by rotating
counterclockwise around the origin of the coordinate
plane.
Plot (5, –2) on a coordinate plane and draw the
terminal side extending from the origin through that
point.
11
5.1.4: Evaluating Trigonometric Functions
Guided Practice: Example 2, continued
The reference angle is the angle the terminal side
makes with the x-axis.
12
5.1.4: Evaluating Trigonometric Functions
Guided Practice: Example 2, continued
Notice that  is nearly 360°, so the reference angle is
in the fourth quadrant.
The magnitude of the x-coordinate is the length of
the adjacent side and the magnitude of the ycoordinate is the length of the opposite side. The
hypotenuse can be found using the Pythagorean
Theorem. Determine the sign of sin  by recalling the
ASTC pattern or by considering the signs of the xand y-coordinates.
13
5.1.4: Evaluating Trigonometric Functions
Guided Practice: Example 2, continued
2. Find the length of the opposite side and
the length of the hypotenuse.
Sine is the ratio of the length of the opposite side to
the length of the hypotenuse; therefore, these two
lengths must be determined.
The length of the opposite side is the magnitude of
the y-coordinate, 2.
14
5.1.4: Evaluating Trigonometric Functions
Guided Practice: Example 2, continued
Since the opposite side length is known to be 2 and
the adjacent side length, 5, can be determined from
the sketch, the hypotenuse can be found by using
the Pythagorean Theorem.
c2 = a2 + b2
Pythagorean Theorem
c2 = (2)2 + (5)2
Substitute 2 for a and 5 for b.
c2 = 4 + 25
Simplify the exponents.
c2 = 29
Add.
c = 29
Take the square root of both sides.
The length of the hypotenuse is
5.1.4: Evaluating Trigonometric Functions
29 units.
15
Guided Practice: Example 2, continued
3. Find sin  .
Now that the lengths of the opposite side and the
hypotenuse are known, substitute these values into
the sine ratio to determine sin  .
sinq =
sinq =
length of opposite side
Sine ratio
length of hypotenuse
( 2)
( 29 )
5.1.4: Evaluating Trigonometric Functions
Substitute 2 for the
opposite side and
29 for the
hypotenuse.
16
Guided Practice: Example 2, continued
sinq =
2 29
Rationalize the
denominator.
( 29)
According to ASTC, in Quadrant IV only the cosine
and secant are positive. The sine is negative.
For a positive angle  in standard position with
a terminal side that passes through the
point (5, –2), sinq =
2 29
( 29)
.
✔
17
5.1.4: Evaluating Trigonometric Functions
Guided Practice: Example 2, continued
18
5.1.4: Evaluating Trigonometric Functions
Guided Practice
Example 4
4
Given cosq = , if  is in Quadrant I, find cot  .
5
19
5.1.4: Evaluating Trigonometric Functions
Guided Practice: Example 4, continued
1. Sketch an angle in Quadrant I, draw the
associated triangle, and label the sides
with the given information.
Cosine is the ratio of the length of the adjacent side to
4
the length of the hypotenuse. Since cosq = , 4 is the
5
length of the adjacent side and 5 is the length of the
hypotenuse.
20
5.1.4: Evaluating Trigonometric Functions
Guided Practice: Example 4, continued
21
5.1.4: Evaluating Trigonometric Functions
Guided Practice: Example 4, continued
2. Use the Pythagorean Theorem to find the
length of the opposite side.
Since the lengths of two sides of the triangle are given,
substitute these values into the Pythagorean Theorem
and solve for the missing side length.
22
5.1.4: Evaluating Trigonometric Functions
Guided Practice: Example 4, continued
c2 = a2 + b2
Pythagorean Theorem
(5)2 = (4)2 + b2
Substitute 5 for c and 4 for a.
25 = 16 + b2
Simplify the exponents.
9 = b2
Subtract 16 from both sides.
3=b
Take the square root of both sides.
The length of the opposite side is 3 units.
23
5.1.4: Evaluating Trigonometric Functions
Guided Practice: Example 4, continued
3. Find the cotangent.
Use the values from the triangle to determine the
cotangent.
cot q =
length of adjacent side
Cotangent ratio
length of opposite side
4)
(
cot q =
(3)
Substitute 4 for the
adjacent side and 3
for the opposite
side.
24
5.1.4: Evaluating Trigonometric Functions
Guided Practice: Example 4, continued
In Quadrant I, all trigonometric ratios are positive,
which coincides with the answer found.
4
Given cosq = , for an angle  in Quadrant I,
5
4
cot q = .
3
✔
5.1.4: Evaluating Trigonometric Functions
25
Guided Practice: Example 4, continued
26
5.1.4: Evaluating Trigonometric Functions