Lesson 4.1

Right Triangle Trigonometry

Lesson Overview:

Chart for Trig value of special angles: 0,30,45,60,90 degrees

Use SOHCAHTOA to solve special right triangles

Use SOHCAHTOA to solve right triangles: side and angles

Trig Values for Special Angles

Use SohCahToa to Solve Special Right Triangle

Steps:

Label sides: opposite, adjacent, and hypotenuse
Set up an equation to find the missing variable
Use the left hand trick to find the special angle
Solve for the missing variable

Real World Application:

Marco has a square garden in his backyard that measure 8 feet long by 8 feet wide. He wants to make a diagonal path through it. He needs to know how long the path would be to get the right amount of rock to make the path.

We already know that it will be a 45-45-90 triangle because both sides of the triangle will be the same length, 8 feet.
What we need to do is find the hypotenuse. We can do that by using our formula for 45-45-90 triangles. (both sides represent a, while the hypotenuse represents a√2.
So, we take one of our sides, and multiply it by the square root of two.
8(√2 ) = 8√2

SAT Prep Question: If the length of the largest side of a right triangle is 10 inches and one of the angles is 60°, what is the length (in inches) of the smallest side?

First, you need to know the measures of the special right triangle, in this case, it is a 30-60-90 triangle, and we are solving for the smallest side.
If the hypotenuse measures 10 inches, then we know that we have to simply divide by two to get the shortest side, giving us 5 in.

Use SohCahToa to Solve Right Triangle Sides

Steps:

Label Sides: Opposite, adjacent, hypotenuse
Set up an equation to solve for unknown variable
Solve for unknown variable

****Make sure your calculator is in degree mode****

Real World Application:

At 57" from the base of a building you need to look up at 55° to see the top of a building. What is the height of the building?

We know that we have one of the angle measurements and the adjacent side length, and we are looking for the opposite side length.
We are going to use tan to solve, using an angle (x), and the adjacent side (a) to find the opposite side (o).
First, we need to set up our equation, tan(x)=opposite/adjacent, then fill in what we know. Now we have tan(55)=opposite/57.
Now we can solve for the opposite side using basic algebraic skills. First getting opposite on its own, which bring us to tan(55) * 57 to give us to give us somewhere around 81.4".

SAT Prep Question: If angle B measures 53° and the hypotenuse of the triangle measures 10 inches, the what is the measurement of the opposite side in inches?

First, we have to find which equation to use, in this case, it will be sin, because we know the hypotenuse and we are looking for the opposite.
All you do is plug into the formula and solve for the opposite, sin(54) = opp/10.
When you solve for the opposite side, you will get somewhere around 8.09 in.

Use SohCahToa to Solve Right Triangle Angles

Steps:

Label Sides: Opposite, adjacent, hypotenuse
Set up the equations to solve for the unknown angle
Move the trig functions (sin, cos, tan) to the other side of the equal sign by using the inverse
Solve for the angle

Real World Application:

A boat is 20 feet away from the shore where the lighthouse is. The lighthouse is 15 feet above the ground. At what angle will the lighthouse need to angle its light to see the boat?

We know the adjacent and the opposite measurements of the triangle. All we are looking for is the angle.
Because we have the adjacent (20 feet) and the opposite (15 feet), we will use tan because of the formula, tan(x)=opposite/adjacent
We just have to plug our numbers into the formula. tan(x)=15/20.
15/20 = 0.75, then we use tan^-1(0.75) to get an angle of 36.87 roughly.

SAT Prep Question:

The first step is to figure out which equation you will be using, in this case, it will be sin, because we know the values of the opposite and the hypotenuse.
Next, you just plug in the known values into the equation, sin(?) = opp/hyp which will turn into sin(?) = 4/9
The last step is to solve for the unknown angle, using inverse sin to solve, so do inverse sin 4/9 to get a number around 26.39

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