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
Slide 2
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
Slide 3
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
Slide 4
Key Concepts, continued The cosecant of = csc = ; The secant of
= sec = ; The cotangent of = cot = ; 4 5.1.4: Evaluating
Trigonometric Functions
Slide 5
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 5.1.4: Evaluating Trigonometric Functions
Slide 6
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
Slide 7
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. 7 5.1.4: Evaluating Trigonometric Functions
Slide 8
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
Slide 9
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
Slide 10
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
Slide 11
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
Slide 12
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
Slide 13
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 y- coordinate 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 x- and y-coordinates. 13 5.1.4:
Evaluating Trigonometric Functions
Slide 14
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
Slide 15
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. The length of the hypotenuse is units. 15
5.1.4: Evaluating Trigonometric Functions c 2 = a 2 + b 2
Pythagorean Theorem c 2 = (2) 2 + (5) 2 Substitute 2 for a and 5
for b. c 2 = 4 + 25Simplify the exponents. c 2 = 29Add. Take the
square root of both sides.
Slide 16
Sine ratio Substitute 2 for the opposite side and for the
hypotenuse. 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 . 16
5.1.4: Evaluating Trigonometric Functions
Slide 17
Guided Practice: Example 2, continued 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),. 17 5.1.4: Evaluating
Trigonometric Functions Rationalize the denominator.
Slide 18
Guided Practice: Example 2, continued 18 5.1.4: Evaluating
Trigonometric Functions
Slide 19
Guided Practice Example 4 Given, if is in Quadrant I, find cot
. 19 5.1.4: Evaluating Trigonometric Functions
Slide 20
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 the length of the hypotenuse. Since, 4 is the
length of the adjacent side and 5 is the length of the hypotenuse.
20 5.1.4: Evaluating Trigonometric Functions
Slide 21
Guided Practice: Example 4, continued 21 5.1.4: Evaluating
Trigonometric Functions
Slide 22
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
Slide 23
Guided Practice: Example 4, continued The length of the
opposite side is 3 units. 23 5.1.4: Evaluating Trigonometric
Functions c 2 = a 2 + b 2 Pythagorean Theorem (5) 2 = (4) 2 + b 2
Substitute 5 for c and 4 for a. 25 = 16 + b 2 Simplify the
exponents. 9 = b 2 Subtract 16 from both sides. 3 = bTake the
square root of both sides.
Slide 24
Guided Practice: Example 4, continued 3.Find the cotangent. Use
the values from the triangle to determine the cotangent. 24 5.1.4:
Evaluating Trigonometric Functions Cotangent ratio Substitute 4 for
the adjacent side and 3 for the opposite side.
Slide 25
Guided Practice: Example 4, continued In Quadrant I, all
trigonometric ratios are positive, which coincides with the answer
found. Given, for an angle in Quadrant I,. 25 5.1.4: Evaluating
Trigonometric Functions
Slide 26
Guided Practice: Example 4, continued 26 5.1.4: Evaluating
Trigonometric Functions