Investigation: Right Triangle Trigonometry
When triangles are not special (30-60-90, 45-45-90), we can only estimate the relationships between the
sides by using something called Trigonometry.
But first, we must be able to identify parts of a triangle.
Now that we can identify the sides, there are three special relationships that we use.
To help use remember which special word goes with which sides, we use an acronym called
SOHCAHTOA
You try some! Find the following trigonometric ratios.
a)
b)
c)
d)
e)
f)
Using these equations allow us to find the length of a missing sides of a right triangle if we know
something about the acute angles. *** Keep in mind, you can still you special right triangle relationships
and Pythagorean Theorem, if needed.***
Example: Find the missing lengths using trigonometric ratios.
a)
b)
c)
Your Turn to Try! Find the Value of x.
d)
e)
f)
Multi-step: Use your answer from one triangle to help you find the x in the other.
Solve each triangle. This means, find the measures of all the angles and all the sides.
1.
2.
3.
Word Problems and Inverse Trigonometry
The inverse of something is when you undo an operation. For Trigonometry, the inverse is used to find
the angle that we started with in a triangle. The notation for inverse is:
Sin-1
Cos-1
Tan-1
So when we are looking for an angle (x is the angle), we find the inverse of the ratio of the sides.
Example:
Find the missing angle using trigonometric inverses.
a)
b)
c)
d)
e)
f)
Word Problems
Angle of Elevation and Angle of Depression
1) A surveyor is standing 115 feet from the base of the Washington Monument. The surveyor measures
the angle of elevation to the top of the monument as 78.3.
How tall is the Washington Monument?
2) A helicopter is hovering above a road at an altitude of 24 m. At a certain time, the
distance between the helicopter and a car on the road is 45.0 m. Calculate the angle from
the car to the helicopter.
Solve the following word problems using the Pythagorean theorem, trigonometric ratios or trigonometric
inverses.
1) Joe is standing a distance of 20 m away the foot of a tower. He is looking up at the tower at a 40
degree angle. What is the height of the tower?
2) A person walks up a ramp to a door that has an angle of 20 from the ground. It has a vertical height of
1.8 m. What is the length of the ramp? How far is the person from the door?
3) A fire department’s longest ladder is 110 feet long, and the safety regulation states
that they can use it for rescues up to 100 feet off the ground. At what safe angle can the ladder be placed
with the ground?