Download Basic Trigonometry - Solving Right Triangles - Arcadia Valley R

MA 10A
Arcadia Valley Career Technology Center
Topic: Basic Trigonometry
Show-Me Standards: MA2, MA4
Tech Math I Credit
Last Update: March 2008
Focus: Solving Right Triangles ( Six
Trig. Functions,)
MO Grade Level Expectations: G1A9,
G1A11, G2A8, G4B10
NCTM Standards: 8D, 10A
OBJECTIVE: The students will be able to find the values of the six basic trigonometric
functions and solve problems involving right triangles.
*Recall from Lesson 7B
RIGHT ANGLES: angles that measure 90-degrees. A right angle is often shown with a small square
drawn in the corner of the angle.
¬
RIGHT TRIANGLE:
c=hypotenuse – by definition, the side
opposite the right angle.
a=leg
b=leg
c 2  a 2  b 2 , or
Pythagorean Theorem:
a  c2  b2
b  c2  a2
, where c is the hypotenuse.
c  a2  b2
OTHER IMPORTANT INFORMATION:
1. The sum of the angles of any triangle is 180-degrees.
2. The sum of the two acute angles of the right triangle is 90-degrees.
3. A triangle without a right angle (an ‘oblique triangle’) can be worked with by first creating two
right triangles. Working from the known values, the two triangles can be solved and results
combined to give the desired angles and sides of the oblique triangle.
opposite
hypotenuse
adjacent
4. The Trigonometric Functions are: cos  
, another way of remembering this
hypotenuse
opposite
tan  
adjacent
Old
Some 
Horse
Another
information is as follows: Caught 
.
Horse
Oats
Taking 
Away
sin  
Now, we are going to look at the other three basic Trigonometric Functions:
cosecant (csc)
secant(sec)
cotangent(cot)
csc θ = hypotenuse
opposite
sec θ = hypotenuse
adjacent
cot θ = adjacent
opposite
-
Notice that the sin, cos, and tan are reciprocals of the csc , sec, and cot respectively.
Therefore, the following are true:
csc θ = _ 1__
sin θ
sec θ =
1__ _
cos θ
cot θ =
1 ___
tan θ
Examples:
If sin θ = 4_ or 0.8 , then find the other five trig. function values.
5
-
we can use the Pythagorean Thm. to find the missing leg. The opposite leg must be 4 and
the hypotenuse is 5 so,
(adj.leg)2 + 42 = 52 and adj. leg = 3
-
cos θ = 3 _
5
or 0.6
tan θ = 4_
3
or 1.3333
csc θ = 5_
4
or 1.25
sec θ = 5_
3
or 1.6667
cot θ = 3_
4
or 0.75
*Recall from Lesson 7B
-
Inverses of the basic trig. functions are used when you know the value of the trig. function
but you would like to know the measure of the angle that goes with it. (symbol for inverse
is -1 )
Examples:
sin-1 (½) = 30°
because the sin 30° = ½
cos-1 (.5667) = 56.2°
Solving a Right Triangle – if you know the measures of any two sides of a right triangle or the
measure of any one side and one of the acute angles, you can find all the missing measures. This is
called solving the right triangle.
A
-
Examples:
c
b
C
a
ABC is a right triangle
B
Ex 1.
If a = 10 “ and Angle B = 35°, then solve the right triangle.
Step 1
tan 35° = b_ , 0.7002 =
10
b_ ,
10
b = 7.0”
Step 2
90° - 35° = 55° ,
Angle A = 55°
Step 3
cos 35° = 10_
c
or
, 0.8192 =
102 + 7.02 = c2
, √ 149 = c
10_
c
,
,
c = 12.2”
c = 12.2”
So,
a = 10”
b = 7.0”
c = 12.2”
Angle A = 55°
Angle B = 35°
Angle C = 90°
Ex 2.
If b = 6 cm and c = 13 cm , then solve the right triangle.
Step 1 a2 + 62 = 132
,
a2 + 36 = 169 , a = √133 , a = 11.5 cm
Step 2 sin-1 6_ = Angle B ,
13
Angle B = 27.5°
Step 3 90° - 27.5° = 62.5° ,
Angle A = 62.5°
So,
a = 11.5 cm
b = 6 cm
c = 13 cm
Angle A = 62.5°
Angle B = 27.5°
Angle C = 90°
Note: There is usually more than one way to solve a right triangle. You may want to go back and
redo the two examples using different steps. The answers should stay the same.
Problems:
1. Find the values for the five other trigonometric functions given sin θ = 8_ .
17
2. Find the values for the five other trigonometric functions given sec θ = 16_
12
3. Solve the right triangle given the following measures:
Angle A = 35° , a = 12 mm
4. Solve the right triangle given the following measures:
a = 4 ft. , b = 7 ft.
5. Solve the right triangle given the following measures:
Angle A = 17° ,
c = 3.2 miles
.
SOLVE:
6. A sightseeing boat is near the base of a waterfall and the captain estimates the angle of elevation
to the top of the falls to be 30°. If the falls are 180 ft. high, how far is the boat from the base of
the falls?
7. A surveyor stands 200 ft. from a bridge pillar. The bridge pillar is 320 ft. high. What is the
surveyor’s angle of sight to the top of the pillar?
8. A tread mill is 48” long and set on a 6° incline. What is the vertical rise of the treadmill?
9. The tailgate of a moving truck sits 3’ above the ground with a ramp out the back at a 15°
incline with the ground. How long is the ramp?
10. A 38’ guide wire runs from the top of a pole to the ground. The wire makes an angle with the
ground of 52°. How far is the pole from the anchored guide wire? How tall is the pole?