Dorchester 2-02-10-2025-Trig Elev/Dep, & Ca

​Use the sine, cosine, and tangent ratios to determine side lengths and angle measures in right triangles in real world situations.

Open Ended

Which one doesn't belong?

Use SOH CAH TOA to select the correct ratios of sine, cosine, and tangent.​

We use these ratios in order to solve missing sides and angles in a right triangle​.

​Trig Ratios

​

Where is trigonometry used in the real world?

* Video Game Design

* Speaker Placement in Concert Venues

* Architecture/Construction

* Criminology

* Military

​Remember the Steps

STEP 1: Label your Sides and the reference angle.
STEP 2: What am I Solving For?
STEP 3: Which trig function do I need? sin cos tan
STEP 4: Write the TRIG EQUATION-YOU CANNOT SKIP THIS STEP
STEP 5: Find the cross product
STEP 6: cross multiply and solve Calculator Time!




STEP 7: round to instructed digit

A. Click on the wrench in the upper right corner. Click on Degrees. SUPER IMPORTANT!

B. Click on Keyboard icon- then FUNCTIONS, then TRIG.

C. Enter the equation

D. Round as instructed

MISSING SIDE

Multiple Choice

True or false. Trigonometric functions work for ALL triangles.

• True
• False

Multiple Choice

From <C, what side is AB?

• Adjacent

• Opposite

• Hypotenuse

Multiple Choice

Which side is the hypotenuse?

• 1
• 2
• 3

Multiple Choice

Which side is opposite angle x?

• 1
• 2
• 3

Multiple Choice

Which side is adjacent to angle x?

• 1
• 2
• 3

Multiple Choice

Choose the correct ratio.

• sin(37)=x/14
• cos(37)=x/14
• tan(37)=x/14
• sin(37)=14/x

Multiple Choice

Which is the correct set up to solve for x?

• $$\frac{\sin24}{1}=\frac{x}{12}$$

  
  

• $$\frac{\cos24}{1}=\frac{x}{12}$$

• $$\frac{\tan24}{1}=\frac{x}{12}$$

• $$\frac{\tan\ 24}{1}=\frac{12}{x}$$

Multiple Choice

Which triangle is correctly labeled?

Math Response

Solve for x, round to the nearest tenth

Type answer here

Correct Answer: 8.1

Deg^°

Rad

Math Response

What is the value of x?

Round to the nearest ten-thousandth place.

Type answer here

Correct Answer: 8.9200

Deg^°

Rad

Open Desmos Calculator
Follow along with the video
​DESMOS - Inverse Trig for Missing Angles.mp4

Multiple Choice

Which of the following would you use to find the measure of an angle?

• Pythagorean theorem
• Sin, Cos, or Tan
• Sin^-1, Cos^-1, or Tan^-1
• Chord-Chord Product Theorem

Multiple Choice

What should be typed into a calculator (in degree mode) to solve for x?

• $$\sin\left(\frac{3}{7}\right)$$

• $$\sin^{-1}\left(\frac{3}{7}\right)$$

• $$7\sin^{-1}\left(3\right)$$

• $$\sin^{-1}\left(\frac{7}{3}\right)$$

Multiple Choice

tan(216)

• 3

• 6

• 0.2

• 0.7

Multiple Choice

Using your calculator round to the nearest hundredth

• 0.10

• 0.11

• 1.4

• 104

Multiple Choice

Using your calculator round to the nearest hundredth

• 0.84

• 0.88

• 8.4

• 84

Multiple Choice

Using your calculator round to the nearest hundredth

• 0.60

• 0.61

• 6.0

• 60

Multiple Choice

Using your calculator round to the nearest hundredth

• 1.15

• 1.14

• 1.2

• 115

Multiple Choice

Using your calculator round to the nearest hundredth

• 0.27

• 2.7

• 0.25

• 27

Multiple Choice

Using your calculator round to the nearest hundredth

• 0.28

• 0.26

• 0.27

• 28

Multiple Choice

Using your calculator round to the nearest hundredth

• 0.83

• 0.82

• 0.84

• 0.8

​Next steps

Solving for a Missing Angle

​Inverse Trig Ratios

​There are two types of notation for inverse functions

REGARDLESS OF NOTATION
WE ALWAYS SAY "inverse" NEVER "arc" OR "to the -1"

• arcsin = sin^-1 we say "inverse sine" or "sine inverse."

• arccos = cos^-1 we say "inverse cosine" or "cosine inverse."

• arctan = tan^-1 we say "inverse tangent" or "tangent inverse."

STEP 1: Label your GIVEN Sides and reference angle.
STEP 2: What am I Solving For?
STEP 3: Which trig function do I need? sin cos tan
STEP 4: Write the TRIG EQUATION - YOU CANNOT SKIP THIS STEP
STEP 5: Input equation into Desmos Calc
STEP 6: Round to instructed digit

​Solving for a Missing Angle

Using trig. to calculate an angle measure
EXAMPLES - Write these down

​Write the equation next to the example.

​​Write the equation next to the example.

Example 1

Draw

Label the given sides outside the triangle: O, H, or A

Write the trig ratio for theta on the upper left corner.

Go to DESMOS Calculator:

https://www.desmos.com/calculator

Write the angle measure inside the triangle.

Example 2

Draw

Label the given sides outside the triangle: O, H, or A

Write the trig ratio for theta on the upper left corner.

Go to DESMOS Calculator:

https://www.desmos.com/calculator

Write the angle measure inside the triangle.

Example 3

Draw

Label the given sides outside the triangle: O, H, or A

Write the trig ratio for theta on the upper left corner.

Go to DESMOS Calculator:

https://www.desmos.com/calculator

Write the angle measure inside the triangle.

Poll

• I can apply the trigonometric ratios to find the missing angles in right triangles.

• I've Got This

• Almost There

• Hmmm--Not Sure

• I Need Your Help

Vocabulary

Vocabulary

​The angle of depression is congruent to the angle of elevation because they are alternate interior angles.

Time for independent practice

Multiple Choice

Use the SIN^-1 formula to find the measure of the missing angle to the nearest whole degree.

• 40 degrees

• 45 degrees

• 50 degrees

• 55 degrees

Multiple Choice

Use the COS^-1 formula to find the measure of the missing angle to the nearest whole degree.

• 48 degrees

• 54 degrees

• 60 degrees

• 66 degrees

Multiple Choice

Use the TAN^-1 formula to find the measure of the missing angle to the nearest whole degree.

• 32 degrees

• 40 degrees

• 51 degrees

• 58 degrees

Multiple Choice

Find the measure of the missing angle to the nearest whole degree.

• 50 degrees

• 48 degrees

• 42 degrees

• 40 degrees

Multiple Choice

Find the measure of the missing angle to the nearest whole degree.

• 28 degrees

• 30 degrees

• 33 degrees

• 57 degrees

Multiple Choice

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

• 66.4 ft

• 1.59 ft
• 74.2 ft
• 86.9 ft

Math Response

Henry is having a picnic with his friends on the beach. When he looks directly out at the ocean, he can see a pier and a lighthouse on the shore. Henry is 150 feet from the shore. The pier is 64° to his left, and the lighthouse is 70° to his right. What is the distance between the pier and the lighthouse?

Round your answer to the nearest foot.

Type answer here

Correct Answer: 720

Deg^°

Rad

Multiple Choice

A hawk sitting on a tree branch spots a mouse on the ground 15 feet from the base of the tree.  The hawk swoops down toward the mouse at an angle of 30 degrees. What is the distance from the tree branch to the mouse? 

• 7.5 ft 
• 15 ft 
• 26 ft 
• 30 ft 

Multiple Choice

Solve for the missing distance.

• 18.9
• 19.5
• 47
• 27.5

Multiple Choice

A yacht is anchored 90 feet offshore from the base of a lighthouse.  The angle of elevation from the boat to the top of the lighthouse is 26 degrees.  The distance between the yacht and the top of the lighthouse is about 100 feet.  

Which of these is closest to the height of the lighthouse? 

• 25 feet
• 44 feet 

• 110 feet 

• 135 feet 

Multiple Choice

Jake is 6 ft. tall. He is standing outside when the angle of elevation of the sun is 28 degrees. How long would his shadow be? 

• 5.3 feet
• 2.8 feet
• 3.2 feet
• 11.3 feet

Multiple Choice

A 10 ft. tree creates a shadow that is 15 ft. long. Find the angle of elevation of the sun. 

• 34 degrees
• 56 degrees
• 72 degrees
• 15 degrees

Multiple Choice

A 13 ft. ladder is leaning against the house. The top of the ladder touches the house at 10 feet above the ground.

How far away is the bottom of the ladder from the house?

• 10 feet

• 7.5 feet

• 8.3 feet

• 3.8 feet

Multiple Choice

A 30 ft. wire stretches from the top of a light pole diagonally to the ground. The light pole is 10 feet high.

What is the angle the wire makes with the ground?

• 66 degrees
• 80 degrees
• 30 degrees
• 19 degrees

Multiple Choice

A new pipeline is being created to shorten the distance it has to travel to get from point A to point B. The original pipeline ran at a 90 angle from point A to point B and was 20 miles on one side and 15 miles on the other. The new pipeline will go straight from A to B.

How much shorter will the new pipeline be?

• 10
• 25
• 175
• 625

Multiple Choice

A 10 ft. tree creates a shadow that is 15 ft. long. Find the angle of elevation of the sun. 

• 34 degrees
• 56 degrees
• 72 degrees
• 15 degrees

Multiple Choice

Which of these equations should be used to determine the angle (x) that the Learning Tower of Pisa makes with the ground?

Click on the image.

• A

• B

• C

• D

Multiple Choice

Jake is 6 ft. tall. He is standing outside when the angle of elevation of the sun is 28 degrees. How long would his shadow be?

• 5.3 feet

• 2.8 feet

• 3.2 feet

• 11.3 feet

Multiple Choice

Which trig function will be used to find the missing angle?

• sin

• cos-1

• tan-1

• sin-1

Multiple Choice

Choose the correct ratio.

• sin (9/20)

• cos (9/20)

• tan (9/20)

• cos (20/9)

Multiple Choice

Choose the correct ratio.

• sin^-1(12/29)
• cos^-1(12/29)
• tan^-1(12/29)
• tan^-1(29/12)

Multiple Choice

Choose the correct ratio.

• sin (x) = 14/29

• cos (x) = 14/29

• tan (x) = 14/29

• tan (x) = 29/14

Multiple Choice

Choose the correct equation.

• sin^-1 X = 41/28

• tan^-1 X = 41/28

• cos^-1 X = 28/41

• sin^-1 X = 28/41

Multiple Choice

Choose the correct ratio.

• sin (x) = 14/29

• cos (x) = 14/29

• tan (x) = 14/29

• tan (x) = 29/14

Multiple Choice

Choose the correct ratio.

• sin (x) = 11/27

• sin (x) = 27/11

• cos (x) = 11/27

• tan (x) = 11/27

Multiple Choice

Choose the correct ratio.

• sin (x) = 6/8

• cos (x) = 8/6

• cos (x) = 6/8

• tan (x) = 6/8

Multiple Choice

Which equation would be best to use to solve for the missing angle?

• $$\tan^{-1}\left(\frac{12}{13}\right)$$  

• $$\sin^{-1}\left(\frac{12}{13}\right)$$  

• $$\cos^{-1}\left(\frac{12}{13}\right)$$  

• $$\tan\left(\frac{12}{13}\right)$$  

Multiple Choice

Solve for the missing angle. Round to the nearest tenth.

• 1 degree

• 22.6 degrees

• 42.7 degrees

• 67.4

Multiple Choice

Which equation would be best for solving for the missing angles?

• $$\sin^{-1}\left(\frac{4}{13}\right)$$  

• $$\cos^{-1}\left(\frac{4}{13}\right)$$  

• $$\tan^{-1}\left(\frac{4}{13}\right)$$  

• $$\tan\left(\frac{4}{13}\right)$$  

Multiple Choice

Find the measure of the indicated angle.  

• $$\sin^{-1}\left(\frac{15}{16}\right)$$

• $$\cos^{-1}\left(\frac{15}{16}\right)$$  

• $$\tan^{-1}\left(\frac{15}{16}\right)$$  

• Cannot be determined

Multiple Choice

How to construct a right triangle to solve a word problem

• i

• ii

• iii

• iv

Draw

Draw an image that shows your confidence level in finding the measures of missing sides and angles.

Make sure your image is school appropriate.