T7.2 Right Triangle Trigonometry Presentation

T7.2 Right Triangle Trigonometry
Chapter 7 The Unit Circle: Sine and Cosine
Functions

Concepts and Objectives
⚫ The objectives for this section are
⚫ Use right triangles to evaluate trigonometric
functions
⚫ Use equal cofunctions of complementary angles
⚫ Use the definitions of trigonometric functions for any
angle
⚫ Use right-triangle trigonometry to solve applied
problems

Trigonometric Ratio Review
⚫ In Geometry, we learned that for any given right triangle,
there are special ratios between the sides.
A
opposite
adjacent
=
opposite
sin
hypotenuse
A
=
adjacent
cos
hypotenuse
A
=
opposite
tan
adjacent
A

Trigonometric Functions
⚫ Consider a circle centered at the origin with radius r:
⚫ The equation for this circle is x2 + y2 = r2
⚫ A point (x, y) on the circle creates a right triangle whose
sides are x, y, and r.
⚫ The trig ratios are now (x, y)
r
x
y

=
sin
y
r

=
cos
x
r

=
tan
y
x


Trigonometric Functions
⚫ There are three other ratios in addition to the three we
already know : cosecant, secant, and cotangent.
⚫ These ratios are the inverses of the original three:
(x, y)
r
x
y

= =
1
csc
sin
r
y


= =
1
sec
cos
r
x


= =
1
cot
tan
x
y



Finding Function Values
⚫ Example: The terminal side of an angle  in standard
position passes through the point (15, 8). Find the
values of the six trigonometric functions of angle .
(15, 8)


8
15
(15, 8)

We know that x = 15 and y = 8, but
we still have to calculate r:
Now, we can calculate the values.
= +
2 2
r x y
= + =
2 2
15 8 17 17

8
15
(15, 8)

17
= =
8
sin
17
y
r

= =
15
cos
17
x
r

= =
8
tan
15
y
x

= =
17
csc
8
r
y

= =
17
sec
15
r
x

= =
15
cot
8
x
y


The Unit Circle
⚫ Angles in standard position whose terminal sides lie on
the x-axis or y-axis (90°, 180°, 270°, etc.) are called
quadrantal angles.
⚫ To find function values of quandrantal angles easily, we
⚫ Notice that at the quadrantal
angle points x and y are either
0, 1, or –1 (r is always 1).
use a circle with a radius of 1, which
is called a unit circle.
90
(0, 1)
(0, –1)
270
180
(–1, 0)
0/360
(1, 0)

Values of Quadrantal Angles
⚫ Example: Find the values of the six trigonometric
functions for an angle of 270°.

At 270°, x = 0, y = –1, r = 1.
−
 = = −
1
sin270 1
1
 = =
0
cos270 0
1
−
 = =
1
tan270 undefined
0
(0, –1)

At 270°, x = 0, y = –1, r = 1.
 = = −
−
1
csc270 1
1
 = =
1
sec270 undefined
0
 = =
−
0
cot270 0
1
(0, –1)

Identifying an Angle’s Quadrant
⚫ To identify the quadrant of an angle given certain
conditions, note the following:
⚫ In the first quadrant, x and y are both positive.
⚫ In QII, x is negative and y is positive.
⚫ In QIII, both are negative.
⚫ In QIV, x is positive and y is
IV
III
II I
(+,+)
(–,+)
(–,–)
negative.
(+,–)

⚫ Example: Identify the quadrant (or possible quadrants)
of an angle  that satisfies the given conditions.
a) sin  > 0, tan  < 0 b) cos  < 0, sec  < 0

⚫ Example: Identify the quadrant (or possible quadrants)
of an angle  that satisfies the given conditions.
a) sin  > 0, tan  < 0 b) cos  < 0, sec  < 0
I, II II, IV
II
II, III II, III
II, III

Classwork
⚫ Algebra & Trigonometry 2e
⚫ 7.2: 10-22 (even); 7.1: 40-48 (even)
⚫ College Algebra 2e
⚫ 9.6: 30-38 (even)
⚫ T7.2 Classwork Check
⚫ Quiz T7.1

T7.2 Right Triangle Trigonometry Presentation

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T7.2 Right Triangle Trigonometry Presentation