Lecture 13 sections 5.1-5.2 angles & right triangles

MATH 107
Sections 5.1
Angles and their Measure

The initial side is always located on the positive-x-axis;
the vertex is always the origin.

3© 2011 Pearson Education, Inc. All rights reserved
ANGLES
An angle in a
rectangular coordinate
system is in standard
position if its vertex is
at the origin and its
initial side is the
positive x-axis.

ANGLES
An angle in standard
position is said to lie
in a quadrant if its
terminal side lies in
that quadrant.

MEASURING ANGLES BY USING DEGREES
An acute angle has measure
between 0° and 90°.
A right angle has measure 90°,
or one-fourth of a revolution.
An obtuse angle has measure
between 90° and 180°.
A straight angle has measure
180°, or half a revolution.

Angle Measure
Acute 0° < θ < 90°
Right 90°
Obtuse 90° < θ < 180°
Straight 180°

EXAMPLE 1 Drawing an Angle in Standard Position
Draw each angle in standard position.
a. 60° b. 135° c. −240° d. 405°
Solution
a. Because 60 = (90),
a 60° angle is of a
90° angle.
3
2
3
2

Solution continued
b. Because 135 = 90 + 45, a 135º angle is a
counterclockwise rotation of 90º, followed by
half a 90º counterclockwise rotation.

Solution continued
c. Because −240 = −180 − 60, a −240º angle is a
clockwise rotation of 180º, followed by a
clockwise rotation of 60º.

Solution continued
d. Because 405 = 360 + 45, a 405º angle is one
complete counterclockwise rotation, followed
by half a 90º counterclockwise rotation.

CONVERTING BETWEEN
DEGREES AND RADIANS
radians
180
degree1
π
=
degrees
180
radian1
π
=
radians
180






=°
π
θθ
degrees
180
radians 





=
π
θθ
Degrees to radians:
Radians to degrees:

EXAMPLE 3 Converting from Degrees to Radians
Convert each angle in degrees to radians.
a. 30° b. 90° c. −225° d. 55°
Solution
a.
b.

EXAMPLE 3 Converting from Degrees to Radians
Solution continued
d.
c.

(a) 30° (b) 120° (c) - 60° (d) 270° (e) 104 °
Answers on next slide.

EXAMPLE 4 Converting from Radians to Degrees
a. radians
3
π
Convert each angle in radians to degrees.
c. 1 radian
Solution
180º 180
a. radians 60º
3 3 3
º
π
π
π  
= × = = ÷
 
180º3 3 3
b. radians 180º 135º
4 4 4π
π π  
− = − × = − = − ÷
 
c. 1 radian 1
180
7 º
º
5 .3
π
= × ≈
radians
4
3
b.
π
−

5
(a) radian (b) radian (c) radians (d) 5 radians
3 2 6
π π π
−
Answers on next slide.
s s

ARC LENGTH FORMULA
s = rθ
Where:
r is the radius of the circle
θ is in radians.

EXAMPLE 6 Finding Arc Length of a Circle
A circle has a radius of 18 inches. Find the
length of the arc intercepted by a central angle
with measure 210º.
Solution

MATH 107
Sections 5.2
Right Triangle Trigonometry

EXAMPLE 1
Finding the Values of Trigonometric
Functions
Find the exact values for the six trigonometric
functions of the angle θ in the figure.
Solution
a2
+ b2
= c2
3( )2
+ 7( )
2
= c2
9 + 7 = c2
16 = c2
4 = c

EXAMPLE 1
Solution continued
Finding the Values of Trigonometric
Functions
sinθ =
opp
hyp
=
3
4
cosθ =
adj
hyp
=
7
4
tanθ =
opp
adj
=
3
7
=
3 7
7
cscθ =
hyp
opp
=
4
3
secθ =
hyp
adj
=
4
7
=
4 7
7
cotθ =
adj
opp
=
7
3
Now, with c = 4, a = 3, and b = , we have7

Examples
Find exact values of 6 trig functions for right triangle
with opposite side of length 4 and hypotenuse of 5

EXAMPLE 3
Solution
Finding the Trigonometric Function Values
for 45°.
Use the figure to find sin 45°,
cos 45°, and tan 45°.
2
2
2
1
hypotenuse
opposite
45sin ===°
2
2
2
1
hypotenuse
adjacent
45cos ===°
1
1
1
adjacent
opposite
45tan ===°

Example
Find the other 5 trig fcts of θ given θ is acute angle of
right triangle and cos θ = 1/3

Common trigonometric values
***Have these memorized, but be able to re-derive them if necessary.***
Also remember the two special right triangles and their ratios of sides:
45-45-90 degree triangle: 1, 1, √2 ratio of sides
30-60-90 degree triangle: 1, √3, 2 ratio of sides

PYTHAGOREAN IDENTITIES
1cossin 22
=+ θθ
θ22
sectan1 =+ θ
θ22
csccot1 =+ θ
The cofunction, reciprocal, quotient, and
Pythagorean identities are called the
Fundamental identities.

APPLICATIONS
Angles that are measured between a line of
sight and a horizontal line occur in many
applications and are called angles of elevation
or angles of depression.
If the line of sight is above the horizontal line,
the angle between these two lines is called the
angle of elevation.
If the line of sight is below the horizontal line,
the angle between the two lines is called the
angle of depression.

EXAMPLE 8
A surveyor wants to measure the height of
Mount Kilimanjaro by using the known height
of a nearby mountain. The nearby location is
at an altitude of 8720 feet, the distance
between that location and Mount
Kilimanjaro’s peak is 4.9941 miles, and the
angle of elevation from the lower location is
23.75º. See the figure on the next slide. Use
this information to find the approximate height
of Mount Kilimanjaro (in feet).
Measuring the Height of Mount Kilimanjaro

EXAMPLE 8 Measuring the Height of Mount Kilimanjaro

Solution
The sum of the side length h and the location
height of 8720 feet gives the approximate
height of Mount Kilimanjaro. Let h be
measured in miles. Use the definition of sin θ,
for θ = 23.75º.
EXAMPLE 8 Measuring the Height of Mount Kilimanjaro
h = (4.9941) sin θ
= (4.9941) sin 23.75°
h ≈ 2.0114
9941.4hypotenuse
opposite
sin
h
==θ

Lecture 13 sections 5.1-5.2 angles & right triangles

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Lecture 13 sections 5.1-5.2 angles & right triangles