Special Right Triangles and Trigonometric Ratios

8.2 Special Right Triangles (45-45-90)

By Lilly Halom

Draw

Find the value of x and y.

Draw

Find the value of x and y.

Draw

Find the value of x and y.

Draw

Find the value of x and y.

Draw

Find the value of x and y.

Draw

Find the value of x and y.

Multiple Choice

Find the missing side lengths.

• a = 4, b = 4

• a = 4, b = 2√2

• a = 2√2,  b = 4

• a = 4, b = 4

Multiple Choice

What is the length of x in this 45-45-90 triangle?

• 4√2

• 4

• 8

• 4√3

Multiple Choice

What is the length of u and v in this 30-60-90 triangle?

• u = 8 v = 4√3

• u = 4√2 v = 8

• u = 16 v = 8

• u = 4√3 v = 8

Multiple Choice

Find a.

• 3

• $$3\sqrt{3}$$

• 6

• $$\sqrt{3}$$

Multiple Choice

Find x.

• 10

• $$10\sqrt{3}$$

• $$10\sqrt{2}$$

• 20

Multiple Choice

Which side is the long leg in this 30-60-90 triangle? (not the hypotenuse)

• 4

• u

• v

Multiple Choice

Find x.

• 10

• √3

• 5

• 10√3

Multiple Choice

Find x.

• 30°

• 60°

• 90°

• 180°

Fill in the Blank

What is the height of the TV? Round to the 10ths place (one decimal place) if necessary.

Multiple Choice

What is the distance from the University to Nancy's house ?

• 4 miles

• 6 miles

• $$4\sqrt{3}$$ miles

• $$12\sqrt{3}$$ miles

Multiple Choice

What type of special triangle is this?

• 45°-45°-90°

• 30°-60°-90°

• Isosceles

• Obtuse

Right Triangle Trigonometry

GOAL: To apply what we know about trigonometry to degree applications.

This does not need to apply to ONLY the special right triangles. It can apply to ANY right triangle.

​

Recall:

SOHCAHTOA​

​

Multiple Select

Focus in on the 70^o angle...which two sides are "identified" in the figure?

• opposite

  (an unknown)

• hypotenuse

  (4 units)

• adjacent

  (no marking)

• None of these

Multiple Choice

So we should have found;

opposite side = ?

hypotenuse = 4

(remember...related to the 70^o angle!!!) Which trig ratio uses these two sides?

• sine (sin)

• cosine (cos)

• tangent (tan)

• None of these

​sin70^o = ?/4

​

Check to be sure your calculator is in DEGREE MODE.

If you need help here - ask Mrs. G.​

​

​So - consider this;

​

Multiple Select

Let's consider angle B (25^o). Which two sides do "b" and "8" represent?

• Opposite

• Adjacent

• Hyptonuse

• All of them

Multiple Choice

Let's focus on these two;

Opposite Side = "b"

Adjacent Side = "8"

(in relationship to angle B)

Which trig ratio uses these two sides?

• sine (sin)

• cosine (cos)

• tangent (tan)

• none of them

Multiple Choice

So now...which of the following equations would hold true?

• $$\tan\left(25\degree\right)=\frac{b}{8}$$  

• $$\tan\left(8\degree\right)=\frac{25}{b}$$  

• $$\tan\left(25\degree\right)=\frac{8}{b}$$  

• $$\tan\left(b\degree\right)=\frac{8}{25}$$  

Multiple Choice

Which of the following best represents the length of side "b?"

• 3.73

• 177.88

• 17.16

• 17.74

Multiple Choice

Choose the correct set up to find the length of the side x.

• $$\sin\left(37\degree\right)=\frac{x}{14}$$  

• $$\cos\left(37\degree\right)=\frac{x}{14}$$  

• $$\tan\left(37\degree\right)=\frac{x}{14}$$  

• None of these are correct

Multiple Choice

Find the value of x.

• 8.43

• 11.18

• 10.55 

• None of these are correct

Multiple Choice

What is the correct set up for the problem to solve for side w?

• $$\sin\left(40\degree\right)=\frac{w}{28}$$  

• $$\cos\left(40\degree\right)=\frac{w}{28}$$  

• $$\tan\left(40\degree\right)=\frac{w}{28}$$  

• $$\sin\left(40\degree\right)=\frac{28}{w}$$  

Multiple Choice

What is the length of side "w?"

• 18.0 

• 21.4

• 23.5

• 43.56 

Multiple Choice

OK, go...Find the length of "x."

• 6.57 cm

• 7 cm

• 7.44 cm

• 26.33 cm

Multiple Choice

BELLWORK: Solve for the misssing distance

• 18.9 meters

• 19.5 meters

• 47.2 meters

• 27.5 meters

Multiple Choice

BELLWORK: Find the measure of the indicated angle.

• 9.39

• 55.2

• 20.2

• 19.4

Trigonometry Applications

Today we'll learn how to find missing distances and angles for word problems, using sine, cosine, and tangent (or their inverses).

How to solve:

1. Read the problem.

2. Draw a picture (right triangle).

3. Identify which part you need to find.

4. Use trig ratios or inverse trig ratios to solve.

Different Angles

Knowing which angle is the reference angle can be tricky.

ANGLE OF ELEVATION: "Looking" up.

ANGLE OF DEPRESSSION: "Looking" down.

Fill in the Blank

EXAMPLE 1: The Bungling Brothers Circus is in town and you are part of the crew that is setting up its enormous tent. The center pole that holds up the tent is 70 feet tall. To keep it upright, a support cable needs to be attached to the top of the pole so that the cable forms a 60 degree angle with the ground.

How long is the cable? Round to the nearest hundredth of a foot.

Fill in the Blank

EXAMPLE 2:

The circus decides to use a different cable that is attached to the ground, 50 feet from the center pole (that is still 70 feet tall). Now what is the angle of elevation, rounded to the nearest degree?

Multiple Choice

Elizabeth is flying a kite, which gets caught in the top of a tree. Use the diagram to estimate the height of the tree.

• 63.6 ft

• 1.59 ft

• 74.2 ft

• 86.9 ft

Multiple Choice

Jordan is stuck at the top of a Ferris wheel. Her mother is standing 38 feet from the base of the wheel watching her. If the angle of elevation from Jordan's mom to Jordan is 73^o, how far off the ground is Jordan?

• 118.2 ft

• 120.9 ft

• 124.3 ft

• 126.5 ft

Multiple Choice

Solve for the missing angle.

• 50.5 degrees

• 12.7 degrees

• 36.9 degrees

• 72.2 degrees

Multiple Choice

You are attacking an enemy castle. Your strategy is to go over the castle wall. You are 1500 meters away from the bottom of the wall. You measure the angle of elevation to the top of the wall at 3^o. How tall is the enemy castle wall?

• 78.6 meters

• 1498.0 meters

• 78.5 meters

• 75.3 meters

Multiple Choice

Kevin leans a 16 foot ladder l against a building. If the angle the ladder makes with the ground is 68°, how high up the building does the ladder reach?

• 39.6

• 6.0

• 14.8

• 13.6