Chapter 5
Trigonometric Functions
Section 1: Angles and Degree Measure
The student will be able to:
• Convert decimal degree measures to degrees, minutes, and seconds and vice versa.
• Find the number of degrees in a given number of rotations.
• Identify angles that are coterminal with a given angle.
An angle may be generated by rotating one of two rays that share a fixed endpoint known
as the vertex. One of the rays is fixed (doesn’t move) along the x axis and is called the
initial side. The other side, which rotates, is called the terminal side.
If an angle is rotated in a counterclockwise direction, it is positive.
If an angle is rotated in a clockwise direction, it is negative.
An angle is in standard position if its initial side is along the x axis.
We usually measure angles in degrees with decimnmals, but its also possible to
measure them in degrees, minutes, and seconds.
Quadrant II
Quadrant 1
Terminal
side of the
angle
Terminal
side of the
angle
Terminal
side of the
Quadrant III
angle
Terminal
side of the
angle
Quadrant IV
To change a decimal degree to degrees, minutes, and seconds:
1. Multiply the decimal portion of the degree measure by 60 to find the number of
minutes
2. Multiply the decimal portion of the minute measure by 60 to find the number of
seconds.
Example:
• Convert 329.125° to degrees, minutes, and seconds
To change from degrees, minutes, and seconds to a decimal:
1°
1. Multiply the minutes by ( )
60′
1°
(
)
3600"
2. Multiply the seconds by
3. Add the degree, answer from #1, and answer from #2 together.
Example:
Convert 35°12’7” to a decimal.
Each rotation on the coordinate plane is 360°. If an angle is 3.7 rotations counterclockwise,
you would multiply 360° by 3.7. The answer would be 1332°.
If an angle is 2.5 rotations clockwise, you would multiply by -360° because it would be a
negative angle. The answer would be -900°.
Ex: Give the angle measure represented by each rotation:
a. 9.5 rotations clockwise
b. 6.75 rotations counterclockwise
Tow angles are coterminal if they share the same terminal side.
45°
405°
Your questions for finding coterminal angles will look like this:
answer
Identify all angles coterminal with each angle. Then find one positive angle and one
negative angle that are coterminal with the angle.
answer
answer
a. 45°
b. 225°
a. 86°
b. 294°
If each angle is in standard position, determine a coterminal angle that is
between 0° and 360°. State the quadrant in which the terminal side lies.
a. 775°
b. -1297°
c. 595°
d. -777°
Reference angles are acute angles formed by the terminal side of an angle and the x axis.
They ARE ALWAYS BETWEEN 0° AND 90°!!!
Copy yellow box on page 280 for reference angle rule.
Quadrant II
Quadrant III
Quadrant 1
Quadrant IV
Find the measure of the reference angle for each angle.
1. 35°
2. 105 °
3. 345 °
4. 225 °
Section 2: Trig Ratios in Right Triangles
The student will be able to find the values of trigonometric ratios for acute
angles of a right triangle.
In a right triangle, one of the angles measures 90°, and the remaining angles are acute
and complementary. The longest side of a right triangle is the hypotenuse and it is
always opposite the right angle. The other sides are either adjacent or opposite.
A
b
c
B
a
C
B is the right angle of this triangle. Side b is the hypotenuse. A
and C are acute angles. Side a is opposite angle A and adjacent to
angle C. Side c is adjacent to angle A and opposite angle C.
Copy the Trigonometric Ratios box on page 285.
Examples:
C
Find the values of the sin, cos, and tan for <B.
Find the values of the sin, cos, and tan for <A.
B
18
33
A
C
Find the values of the sin, cos, and tan for <A.
8
B
Find the values of the sin, cos, and tan for <C.
Copy the purple and white box on page 287.
15
A
Section 3: Trig Ratios on the Unit Circle
Students will be able to:
• find the values of trig ratios on the unit circle
• Find the vlues of trig functions of an angle in standard position given a point on its
terminal side.
A unit circle is a circle of radius 1 and it’s center is at the origin of a coordinate plane.
WHAT TO KNOW ABOUT THE UNIT CIRCLE:
•
•
•
•
The radius is ALWAYS 1!!!
The angle we find is always located at the origin.
𝑠𝑖𝑛𝜃 = 𝑦
𝑐𝑜𝑠𝜃 = 𝑥
Examples:
Use the unit circle to find each value:
1. Cos (180°)
2. Sin (-90°)
3. tan (90°)
Use the unit circle to find the values of sin, cos, and tan for a 210° angle.
1. Find the reference angle for a 210° angle.
2. Find the sin, cos, and tan for that reference angle. (box from yesterday’s lesson on
page 287).
3. Change the sign of these trig ratios to reflect what quadrant you are in.
Find the values of the trig functions for an angle 𝜃 in standard position if a point with the
coordinates (5, -12) lies on its terminal side.
Find the values of the trig functions for an angle 𝜃 in standard position if a point with the
coordinates (-15, 20) lies on its terminal side.
Suppose 𝜃 is an angle in standard position whose terminal side lies in Quadrant IV. If
−2
tan𝜃 is , find sin and cos.
5
Suppose 𝜃 is an angle in standard position whose terminal side lies in Quadrant III. If
−4
sin𝜃 is , find tan and cos.
5
Sec 4: Applying Trig Ratios
The student will be able to use trigonometry to find measures of the sides of right
triangles.
q
p
If 𝑃 = 35° and r = 14, find q.
r
The circus has arrived and the roustabouts must put up the main tent in a field near
town. A tab is located on the side of the tent 40 feet above the ground. A rope is tied
to the tent at this point and then the rope is placed around a stake on the ground. If
the angle that the rope makes with the ground is 50° 15′, how long is the rope? What
is the distance between the bottom of the tent and the stake?
If J = 50°, and j=12, find r.
G
R
50°
J
The chair lift at a ski resort rises at an angle of 20.75°and attains a vertical height of
1200 feet.
a. How far does the chair lift travel up the side of the mountain?
b. A film crew in a helicopter records an overhead view of a skier’s downhill run from
where she gets off the chair lift at the top to where she gets back on the chair lift for
her next run. If the helicopter follows a level flight path, what is the length of that
path?
The angle of elevation is the angle between a horizontal line and the line
of sight from an observer to an object at a higher level.
The angle of depression is the angle between a horizontal line and the
line of sight from the observer to an object at a lower level.
The angle of elevation to the top of a tree is 30° from a point 28 feet away from the
foot of the tree. Find the height of the tree rounded to the nearest feet.
A man on the deck of a ship is 15 ft above sea level. He observes that
the angle of elevation of the top of a cliff is 70° and the angle of
depression of its base at sea level is 50°. Find the height of the cliff and
its distance from the ship.
From the top of a spire of height 50 ft, the angles of depression of
two cars on a straight road at the same level as that of the base of
the spire and on the same side of it are 25° and 40°. Calculate the the
distance between the two cars.
A regular hexagon is inscribed in a circle with diameter 26.6 cm. Find the apothem of the
hexagon.
• An apothem is the measure of a line segment from the center of the polygon to the
midpoint of one of its sides.
• A hexagon is a six-sided figure.
Sec 5: Solving Right Triangles
The student will be able to:
• Evaluate inverse trig functions
• Find missing angle measurements
• Solve right triangles
When you know a trig function, but you don’t know the angle, you can use inverse trig
to find the angle. Inverse trig is found on your calculator by pressing the 2nd key before
the trig key. Trigs and their inverses “undo” each other.
Example:
2
𝑠𝑖𝑛𝑥 =
2
3
𝑐𝑜𝑠𝑥 =
2
𝑡𝑎𝑛𝑥 = 3
D
Find angle E.
3
E
7
tan(𝑐𝑜𝑠
F
−14
)
2
cos(arcos )
5
5
Many cities place restrictions on the height and placement of skyscrapers in order to protect
residents from completely shaded streets. If a 100 foot building casts an 88 foot shadow,
what is the angle of elevation of the sun?
Solve each triangle described, given the triangle below. Round
to the nearest tenth. (Find all the missing parts)
K
a. K = 40°, k = 26
J
L
b. j = 65, l = 55
Sec 6: Law of Sines
The student will be able to:
a. Solve triangles by using the law of sines. If the measures of two angles and a side
are given.
b. Find the area of a triangle if the measures of two sides and the included angle or the
measures of two angles and a side are given.
Law of Sines:
𝑎
𝑏
𝑐
=
=
sin 𝐴
sin 𝐵 sin 𝐶
*** We use this when we know two angles and a nonincluded side.
*** Note: these are NOT right triangles!
Examples:
LMN if 𝐿 = 29°, 𝑀 = 112°, 𝑎𝑛𝑑 𝑙 = 22.
Solve
Solve
ABC if m∠A = 70°, c = 26, a = 25
Area of triangles that aren’t right triangles:
1
𝐾 = 𝑏𝑐𝑠𝑖𝑛𝐴
2
1
2
𝐾 = 𝑎𝑏𝑠𝑖𝑛C
1
2
𝐾 = 𝑎𝑐𝑠𝑖𝑛B
Examples:
Find the area of
ABC if b = 21.2, c = 16.5, and A = 25°.
Find the area of
ABC if a = 6.8, b = 9.3, and C = 57°.
A regular octagon is inscribed in a circle with radius of 5 feet. Find the area of the octagon.
Sec 8: Law of Cosines:
The student will be able to:
a. Solve triangles by using the law of cosines
b. Find the area of triangles if the measures of the three sides are given.
Law of Cosines:
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑏𝑐 cos 𝐵
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑏𝑐 cos 𝐶
Solve the triangle:
A = 39.4°, 𝑏 = 12, 𝑐 = 14
a = 19, b = 24.3, c = 21.8
Suppose you want to fence a triangular lot as shown below. If two sides measure 84
feet and 78 feet and the angle between the two sides is 102°, what is the length of the
fence to the nearest foot?
Find the area of the triangle ABC if a = 24, b = 52, and c = 39.