Pythagorean Theorem and Soh Cah Toa

Right Triangle Ratios

First we will review. You have studied the following concepts so far in this trigonmetry unit:

1) Pythagorean Theorem

2) Pythagorean Triples

3) 45-45-90 triangles

4) 30-60-90 triangles

PYTHAGOREAN TRIPLES

Triples are sets of three side lengths of a RIGHT triangle. If you memorize them, it makes finding the missing side lengths of RIGHT triangles super easy.

Multiple Choice

Find the value of x.

• 36

• 12

• 16

• 25

Multiple Choice

Find the value of x.

• 15

• 16

• 18

• 17

Multiple Choice

What is the hypotenuse if two sides are 6 and 8?

• 10

• 12

• 13

• not enough information

​RULES FOR 30-60-90 TRIANGLES

EXAMPLE:

short leg x 2 = hypotenuse

short leg x square root 3 = long leg

Multiple Choice

Find x.

• 10

• $$10\sqrt{3}$$

• $$10\sqrt{2}$$

• 20

Multiple Choice

Find x.

• 22

• $$11\sqrt{3}$$

• $$11\sqrt{2}$$

• 11

​RULES FOR 45-45-90 TRIANGLES

EXAMPLE:

short leg x sq. root of 2 = hypotenuse

Multiple Choice

What is the length of y in this picture?

• 45

• 5√2

• 90

• 5

Multiple Choice

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

• 4√2

• 4

• 8

• 4√3

Multiple Choice

Use the 45-45-90 theorem to solve for the hypotenuse.

• 16

• 8

• 8√2

• √16

Classifying Side Lengths

To successfully learn section 11-3, you need to first understand how to label the side lengths of a right triangle. We will use three words:

ADJACENT

OPOSITE

HYPOTENUSE.

Multiple Choice

The length across from the right angle in a triangle is called the ...........

• cosine

• tangent

• hypotenuse

• sine

Multiple Choice

What is the length of the hypotenuse?

• 16

• 30

• 34

Multiple Choice

What is the hypotenuse?

• EF

• DF

• DE

Multiple Choice

What is the opposite side to <D?

• EF

• DF

• DE

Multiple Choice

What is the green side labelled as using the starred angle as a starting point?

• adjacent

• opposite

• hypotenuse

Multiple Choice

What is the opposite side to <E?

• EF

• DF

• DE

Multiple Choice

What is the opposite side to <D

• EF

• DF

• DE

Multiple Choice

What side length is ADJACENT to angle X?

• ZY

• YX

• ZX

Multiple Choice

What side length is ADJACENT to angle Z?

• ZY

• YX

• ZX

Multiple Choice

What is the green side labelled as from the starred angle?

• adjacent

• opposite

• hypotenuse

Multiple Choice

What is the adjacent side to <D?

• EF

• DF

• DE

Now you are going to write ratios (fractions) using adjacent, opposite, and hypotenuse. These ratios are called:

• SINE

• COSINE

• TANGENT
  

sine - pronounced "sign"

Ex. In the triangle, 8 is opposite <B and the hypotenuse is 17.

So, the SINE of <B = 8/17

Ex. In the triangle, 15 is opposite <A and the hypotenuse is 17.

So, the SINE of <B = 15/17

Please note that "sine" is usually abbreviated "sin," pronounced "sign."

So if I ask you to find the sin A, I'm asking you for the ratio of the side ADJACENT to angle A divided by the hypotenuse.

Cool?

Multiple Choice

In the figure to the left, the sine of C is....

• 32/40

• 32/24

• 24/40

• 24/32

Multiple Choice

In the figure to the left, the sine of A is....

• 32/40

• 32/24

• 24/40

• 24/32

cosine - pronounced "co-sign"

Ex. In the triangle, 8 is adjacent to <A and the hypotenuse is 17.

So, the COSINE of <A = 8/17

Ex. In the triangle, 15 is adjacent to <B and the hypotenuse is 17.

So, the COSINE of <B = 15/17

Please note that "cosine" is usually abbreviated "cos.

So if I ask you to find the cos A, I'm asking you for the ratio of the side ADJACENT to angle A divided by the hypotenuse.

Cool?

Multiple Choice

• $$\frac{21}{29}$$

• $$\frac{21}{20}$$

• $$\frac{29}{20}$$

• $$\frac{20}{29}$$

Multiple Choice

Find the trig value. REDUCE the ratio before selecting your answer.

• A

• B

• C

• D

Multiple Choice

Find the trig value. Reduce your ratio before selecting your answer.

• A

• B

• C

• D

Multiple Choice

Find the trig value.

• A

• B

• C

• D

Multiple Choice

Find the trig value.

• A

• B

• C

• D

Multiple Choice

Find the trig value.

• A

• B

• C

• D

TANGENT is the ratio of the opposite side length of an angle divided by the adjacent side length.

​TANGENT is abbreviated as "tan."

Multiple Choice

What is the tangent ratio for Angle A?

(opposite/adjacent)

• BC/AC (BC over AC)

• AB/BC (AB over BC)

• AC/BC (AC over BC)

Multiple Choice

• 27/36

• 27/45

• 45/36

• 45/27

Multiple Choice

• 32/40

• 40/24

• 32/24

• 24/32