Notes 7.1 – Properties of Right Triangles
NAME _____________________________
Block _______ Date __________________
Geometry Review:
 Three angles of a triangle have a sum of __________
 The acute angles of a right triangle have a sum of ______. Therefore, they are ___________________ angles
 The longest side of a right triangle is the _______________________
 Pythagorean’s Theorem is: _______________________
 To find the missing sides or angles of a right triangle, use (trig acronym) _____________________
Warm up: Using the “Geometry Review” as an aid, find each of the following without a calculator -
B
1.
B
2.
42o
20
16
C
A
C
b
A
length of side b = ________
Co-function identities:



sin   x   cos x
2

measure of angle A: ________

cos   x   sin x
2



tan   x   cot x
2



csc   x   sec x
2

Think of it this way…. Since the acute angles of a right triangle have a sum of 90o,
A
And
sin(A) = _______
=
cos(B) = ________
b
That means,


sec   x   csc x
2

c
sin(90 – B) = sin(A) and cos(90 – A) = cos(B)
C
a
B
Think of some examples on the unit circle, where sinA = cosB, and A and B are complementary angles.
1. Use the Co-function Identities to complete each statement.
A.) sin 56  cos_____
 
  sin______ (Is this true? We know these unit circle values)!
3
B.) cos 
“Solving” a Triangle: calculate all missing angles and sides
NOTE: The 3 angles of a right triangle are usually denoted by letters A, B, & C (where C is the right angle)
and the length of the sides opposite these angles by the letters a, b, and c
Sometimes the angles are denoted by  ,  , and  .
2. Solve the right triangle in which mB = 65 and a = 14.
*Do NOT use rounded off versions to compute missing sides or angles.*
*Use given information wherever possible
3. Solve the right triangle, given b = 3 and  = 40º.
mA  ________ a  _________
mB  ________ b  _________
mC  ________ c  _________
a = ________ a = _________
b = ________ b = _________
g = ________ c = _________
Angle of Elevation – denotes the angle from the horizontal (often the ground) upward to an object.
Angle of Depression – denotes the angle from the horizontal (typically an imaginary line midair) down to an object
Example: A bird in a tree spots a worm on the ground. The worm sees the bird in a tree. Draw a right triangle and label the
angle of elevation and the angle of depression.
The two angles are _____________________, because
they are ____________________________________.
4. A person standing at the top of a 40-foot tall building
sights a friend on the ground. The angle of depression is
52. How far is the friend from the base of the building?
5. At a point 100 feet from the base of a building, the angle
of elevation to the bottom of a smoke stack is 37, the angle
of elevation to the top is 42. Find the height of the smoke
stack alone.
6. Find the angle of ascent (angle of elevation) for a plane that flies at 275 ft/sec for one minute, if its altitude increases
9000 feet.
Right Triangle Trigonometry Practice Problems:
Name: ________________________
Date: ______________ Block: ____
1. Find the exact value of the six trigonometric ratios of the angle θ. Leave answers in fraction form.
a.]
b.]
10
5
5

θ
12
_______
sin  _______ csc
_______
sin  _______ csc
_______
cos _______ sec
_______
cos _______ sec
_______
tan  _______ cot 
_______
tan  _______ cot 
Find the exact value of the following expressions using co-function and other trigonometric identities
To start, list the pairs of co-functions: _________________, _________________, and _________________
2. sin38° - cos52°
3. tan12° - cot78°
6. 1 – cos2 20° – cos2 70°
*Hint: Use Pythagorean Identity first
4.
cos10°
sin80°
cos 70°
7. tan 20° –
cos20°
*Hint: Change cos or sin, use quotient ID.
10. sec 55
9. cos62°-sin28°
csc 35
12. csc    sec  2   cot    tan  2 
 


 


 10 
 5 
 10 
 5 
æ ö
æ ö
13. cos ç 3p ÷ - sin ç p ÷
è 8 ø
è8ø
5. cos 40°
sin 50°
8. cos 35° sin 55° + sin 35° cos 55°
*Hint: choose one to change into its cofunction. (Can’t you do this another way?)
11. tan 2  48  csc2  42
*Hint: Change one to its co-function,
then use Pythagorean ID
æ 7p ö
sec ç ÷
è 12 ø
14. csc æç 5 ö÷
è 12 ø
Using the right triangle shown to the right and the given information in each problem to find the following:
15. b = 5, β = 20°; find a, c, and α.
α
c
b
β
a
16. b = 4, α = 25°; find a, c, and β.
17. a = 2, c = 5; find b, α, and β.
18. A 22-foot extension ladder leaning against a building makes a 70° angle with the ground. How far up the building
does the ladder touch?
19. The angle of elevation of the sun is 35.1° at the instant it casts the Washington Monument’s shadow 789 feet long.
Find the height of the monument.
20. You are in a blimp, 500 feet in the air. In the distance, you can see Soldier Field off to your right and Adler
Planetarium off to your left. The angle of depression from your blimp to the stadium is 32° and to the planetarium is 23°.
Find the distance between Soldier Field and Adler Planetarium.
a.) Sketch the scenario below:
b.) Focus on the right triangle for Soldier Field
c.) Focus on the right triangle for Adler Planetarium.
Which part of each triangle do we care about? How can we find the total distance from Soldier Field to Adler Planetarium?