Unit 7 Lesson 7.2: Trigonometric Ratios

Unit 7 Lesson 7.2: Trigonometric Ratios

MT: Solving Applied Problems & Modeling in Geometry

SOHCAHTOA notes

​Trigonometric Ratios

SOHCAHTOA notes

​Trigonometric Ratios

• Wh​en using trig functions, you must begin at an angle (do NOT use the 90° angle).

• If we look at angle Θ​, the opposite side sits ACROSS from it and the adjacent side is DIRECTLY NEXT TO it.

• The hypotenuse will ALWAYS stay the same. ​

SOHCAHTOA example

​Trigonometric Ratios

​After you plug in the numbers for the ratios, you can use your calculator to simplify it to a decimal.

SOHCAHTOA example

​Trigonometric Ratios

​Notice that since we switched to ∠B:

• The hypotenuse did not change.

• The opposite and adjacent sides swapped. ​

SOHCAHTOA example

​Trigonometric Ratios

Key Points:

• sin A = cos B

• sin B = cos A

• tan A ×​ tan B = 1

​

These must ALWAYS be true, and you can use these to check that your work is correct!

Multiple Choice

Find the sine, cosine, and tangent of $$\angle R$$ .

• $$\sin R=0.92$$  

  $$\cos R=0.38$$  

  $$\tan R=2.4$$  

• $$\sin R=0.38$$  

  $$\cos R=0.92$$  

  $$\tan R=0.42$$  

• $$\sin R=2.4$$  

  $$\cos R=0.38$$  

  $$\tan R=0.92$$  

Multiple Choice

Find the sine, cosine, and tangent of $$\angle D$$ .

• $$\sin D=1.875$$  

  $$\cos D=0.53$$  

  $$\tan D=0.88$$  

• $$\sin D=0.88$$  

  $$\cos D=0.47$$  

  $$\tan D=1.875$$  

• $$\sin D=0.47$$  

  $$\cos D=0.88$$  

  $$\tan D=0.53$$  

Finding Length notes

When we can't use Pythagorean Theorem to calculate missing sides, we can use a trig ratio instead. ​

​Step 1: Determine which trig ratio you need.

Step 2: Set up the ratio using the pieces from the triangle.

Step 3: Solve for x. ​

Finding Length example

Step 1: Determine which trig ratio you need.

To choose the right ratio, start at the angle (76°​).

• The side x lies OPPOSITE of 76°​.

• The side 6 ft lies ADJACENT to 76°​.

​

The ratio that uses OPPOSITE & ADJACENT is TANGENT. ​

Finding Length example

Step 2: Set up the ratio using the pieces of the triangle.

Finding Length example

Step 3: Solve for x.

Multiple Choice

Find the length of x.

• 148

• 63.2

• 107.5

• 54.3

Multiple Choice

A 12-ft ramp is installed alongside some steps to provide wheelchair access to a library. The ramp makes an angle of 11° with the ground. Find the height of the ramp, x.

• 10.8 ft

• 2.1 ft

• 11.8 ft

• 2.3 ft

Multiple Choice

A 12-ft ramp is installed alongside some steps to provide wheelchair access to a library. The ramp makes an angle of 11° with the ground. Find the distance from the ramp to the wall, y.

• 10.8 ft

• 2.1 ft

• 11.8 ft

• 2.3 ft

Inverse Trig Ratios notes

Every function has an inverse (or "opposite"). The inverse of trigonometric ratios will actually help us calculate missing angles rather than missing sides.

​

​

​

Basically what happens is this: ​

Inverse Trig Ratios notes

​Inverse Trig Ratios:

Inverse Trig Ratios example

Given ΔABC, find ∠A to the nearest whole degree.​

​

Step 1: Determine which trig ratio you need.

Step 2: Set up the ratio like normal.

Step 3: Switch the angle and the ratio.

Step 4: Solve for the missing angle. ​

Inverse Trig Ratios example

Step 1: Determine which trig ratio you need. ​

​To choose the right ratio, start at ∠A:

• The side 19 in. is OPPOSITE of ∠​A.

• The side 36 in. is​ ADJACENT to ∠​A.

​

The ratio that uses OPPOSITE & ADJACENT is TANGENT. ​

Inverse Trig Ratios example

Step 2: Set up the trig ratio like normal. ​

Inverse Trig Ratios example

Step 3: Switch the angle and the ratio. ​

Inverse Trig Ratios example

Step 4: Solve for the missing angle. ​

Inverse Trig Ratios example

Step 4: Solve for the missing angle. ​

Multiple Choice

Find the measure of $$\angle J$$ . Round your answer to the nearest WHOLE NUMBER. 

• $$30\degree$$  

• $$26\degree$$  

• $$60\degree$$  

• $$42\degree$$  

Multiple Choice

Find the measure of $$\angle K$$ . Round your answer to the nearest WHOLE NUMBER. 

• $$60\degree$$  

• $$29\degree$$  

• $$64\degree$$  

• $$26\degree$$  

Multiple Choice

Find the measure of $$\angle Z$$ . Round your answer to the nearest WHOLE NUMBER. 

• $$37\degree$$  

• $$53\degree$$  

• $$31\degree$$  

• $$59\degree$$  

Multiple Choice

Find the measure of $$\angle Y$$ . Round your answer to the nearest WHOLE NUMBER.

• $$37\degree$$  

• $$53\degree$$  

• $$31\degree$$  

• $$59\degree$$