Ch 2 Acute Angles and Right Triangles
2.1 Trigonometric Functions of Acute Angles
“SOHCAHTOA”
sin  
y
opposite
r hypotenuse

csc   
r hypotenuse
y
opposite
x
adjacent
r hypotenuse
cos   
sec   
r hypotenuse
x
adjacent
y opposite
tan   
x adjacent
x adjacent
cot   
y opposite
Find the sine, cosine, and tangent values for
angles A and B.
Find the csc, sec, and cot of angles A and B.
a
sin A   cos B
c
a
tan A   cot B
b
Angle B  90  A
c
sec A   csc B
b
Cofunction Identities
sin A  cos90  A cos A  sin 90  A
sec A  csc 90  A
csc A  sec 90  A
tan A  cot 90  A
cot A  tan 90  A
Write each function in terms of cofunction
cos 52°
tan 71°
sec 24°
Solve for θ
cos  4  sin 3  2
tan 2  18  cot   18
Increasing/Decreasing Functions
As angle A increases, y increases and x decreases
sinA
cosA
tanA
cscA
secA
cotA
sin 21  sin 18 ?
sec 56  sec 49 ?
Special Angles
Find the values of the 6 trig fcns of 30°
hypotenuse=2 opposite=1 adjacent= 3
Find the values of the 6 trig fcns of 60°
hypotenuse=2 opposite= 3 adjacent=1
Find the values of the 6 trig fcns of 45°
hypotenuse=2 opposite= 2
adjacent= 2
2.2 Trigonometric Functions of Non-Acute Angles
A reference angle for an angle θ is the
positive acute angle made by the terminal
side of angle θ and the x-axis.
Find the reference angle for 218°
Find the reference angle for 1387°
(A common error is to find the reference
angle by using the terminal side of θ and the
y-axis.) use the x-axis.
Find the values for the six trig fcns for 210°
x=
y=
r=
now find values for 30°
Finding Trigonometric Function Values For
Any Nonquadrantal Angle θ
1. If θ > 360°, or if θ < 0°, find a coterminal
angle by adding or subtracting 360° as
many times as needed to get an angle
greater than 0° but less than 360°.
2. Find the reference angle θ′.
3. Find the trigonometric function values
for reference angle θ′.
4. Determine the correct signs for the values
found in Step 3. This gives the values of
the trigonometric functions for angle θ.
Find values using reference angles
Find the value of cos 240
Find the value of tan 675°
Find the value of cos 780
Find the value of cot  405
Evaluate cos120  2 sin 2 60  tan 2 30
Find all values of θ, if θ is in the interval
2
0,360and cos    .
2
2.3 Finding Trigonometric Function Values
Using a Calculator
(The calculator must be set in degree mode.)
Approximate sin 49°12’
Approximate sec 97.977°
Approximate
1
cot 51.4283
Approximate sin  246
Use inverse trig fcn to find angles.
sin θ=0.96770915
sec θ=1.0545829
When a vehicle travels uphill or downhill, it
faces a resistance due to gravity. Grade
resistance force is modeled by F  W sin  .
Calculate F for a 2500-lb car on a 2.5°
uphill grade.
Calculate F for a 5000-lb truck on a 6.1°
downhill grade.
Calculate F for θ=0° and θ=90°.
2.4 Solving Right Triangles
Significant Digits
A significant digit is a digit obtained by
actual measurement.
408
21.5
18.00
6.700
0.0025
0.09810
7300
To solve a triangle means to find the measures
of all the angles and sides of the triangle.
Solving an Applied Trigonometry Problem
1. Draw a sketch, and label it with the given
information. Label the quantity to be
found with a variable.
2. Use the sketch to write an equation
relating the given quantities to the
variable.
3. Solve the equation, and check that your
answer makes sense.
When Pat stands 123 feet from a flagpole,
the angle of elevation to the top of the
flagpole is 26°40’. If her eyes are 5.30
feet above the ground, find the height of
the flagpole.
From the top of a 210-ft cliff, David sees
that a lighthouse 430’ offshore. Find the
angle of depression from the top of the
cliff to the base of the lighthouse.
2.5 Further Applications of Right Triangles
Bearing
When a single angle is given, it is
understood that the bearing is measured
in a clockwise direction from due north.
Method 1
Radar stations A and B are on an east-west
line, 3.7 km apart. Station A detects a
plane at C, on a bearing of 61°. Station B
simultaneously detects the same plane, on
a bearing of 331°.
Find the distance from A to C.
The second method for expressing bearing
starts with a north-south line and uses an
acute angle to show the direction, either east
or west, from this line.
Method 2
A ship leaves port and sails on a bearing of
N 47° E for 3.5 hr. It then turns and sails
on a bearing of S 43° E for 4.0 hr. If the
ship’s rate of speed is 22 knots (nautical
miles per hour), find the distance that the
ship is from port.
Using trigonometry to measure distance
The subtense bar method is a method that
surveyors use to determine a small
distance d between two points P and Q.
The subtense bar with length b is centered
at Q and situated perpendicular to the line
of sight between P and Q. Angle θ is
measured, then the distance d can be
determined.
Find d when   123'12" and b  2.0000 cm.
How much change would there be if d were
1” larger?
Angles of Elevation
Francisco needs to know the height of a tree.
From a given point on the ground, he finds
that the angle of elevation to the top of the
tree is 36.7°. He then moves back 50 ft.
From the second point, the angle of
elevation to the top of the tree is 22.2°. Find
the height of the tree to the nearest foot.