1B. Triangles, Special Properties and the Sine Buddies

Question 1

What is the Pythagorean Theorem?

Answer 1

a2 + b2 = c2

Question 2

What is the Converse of the Pythagorean Theorem?

Answer 2

It states that if the square of the longest side of a triangle is equal to the sum of the squares of the smaller sides than the triangle is a right triangle.

Question 3

What are the two different types of Special Right Triangles?

Answer 3

1. 45°- 45°- 90° Triangle
2. 30°- 60°- 90° Triangle

Question 4

What is so cool about the sides of a 45°- 45°- 90° Triangle?

Answer 4

Since it is basically an Isosceles Right Triangle its sides are the same length ,’:)

Question 5

What is the ratio for the side lengths of a 45°- 45°- 90° Triangle?

Answer 5

1:1:√2
same same ⋅√2
hypotenuse = leg ⋅ √2

Question 6

How do you find the length of the sides of a 45°- 45°- 90° Triangle?

Answer 6

• If given one leg, that is your answer. (The legs of an isosceles triangle will always be congruent.)
• If given the hypotenuse, use h = l ⋅ √2

Question 7

What do you do when the hypotenuse of a 45°- 45°- 90° Triangle is a whole number?

Answer 7

You will have to rationalize the denominator.
Ex: 12 = x ⋅ √2
12/√2. (don’t stop here.)
12/√2 ⋅ (√2/√2) (multiply by “one”)
12√2/2 -> 6√2 Boom. Answer

Question 8

What is the ratio for the side lengths of a 30°- 60°- 90° Triangle?

Answer 8

√3:1:2
Equations to use:
hypotenuse = 2 ⋅ shorter leg
longer leg = √3 ⋅ shorter leg

Question 9

The altitude of a right triangle is…

Answer 9

equal to the geometric mean of the hypotenuse segments
- CD2 = AD ⋅ DB
- CD = √AD ⋅ DB

Question 10

What is the bigger brother of the Triangle Sum Theorem?

Answer 10

• The two acute triangles in a right triangle are complimentary, meaning that they equal 90° This is because the third angle will always be 90° and all the angles in a triangle will always equal 180°. ‘,:)

Question 10

The leg in a right triangle is…

Answer 10

equal to the geometric mean of the hypotenuse and the length of the hypotenuse segment adjacent to the leg.
- AC = √AD ⋅ AB
- BC = √AB ⋅ DB

Question 11

What is Theta?

Answer 11

θ
Basically the referenced angle.

Question 12

What are the three Trigonometric Ratios?

Answer 12

1. sin(θ) = O/H (Sine)
2. cos(θ) = A/H (Cosine)
3. tan(θ) = O/A (Tangent)
   Named after their referenced angle.

Question 13

What is Sine?

Answer 13

Sine = Opposite/Hypotenuse
The sine of angle θ is the ratio of the side opposite to the angle and the hypotenuse of the right angle.

Question 14

What is Cosine?

Answer 14

Cosine = Adjacent/Hypotenuse
The cosine of angle θ is the ratio of the side adjacent side to the hypotenuse of the right angle.

Question 15

What is Tangent?

Answer 15

Tangent = opposite/adjacent
The tangent of angle θ is the ratio of the side opposite to the angle over the side adjacent to the angle.

Question 16

How to find missing lengths of a Triangle?

Answer 16

1. Use the Law of Sines
2. If right, use the Pythagorean Theorem
3. Pray to Jesus for Divine Enlightenment

Question 17

What is arcsine?

Answer 17

Basically it is the inverse sine function and it gives the measurement of the referenced angle using the lengths of the hypotenuse and the opposite side.

Question 17

What are the Inverse Trigonometric Ratios?

Answer 17

1. sin-1(O/H) = arcsine
2. cos-1(A/H) = arccosine
3. tan-1(O/A) = arctangent
   😭

Question 18

What is arccosine?

Answer 18

The inverse cosine function that allows you to find the angle measurement of θ using the ratio of lengths of the adjacent side and the hypotenuse.

Question 19

What is arctangent?

Answer 19

The inverse tangent function that gives the angle measurement for θ using the lengths of the opposite side and the adjacent side.

Question 20

How to find the angle of θ using your given?

Answer 20

1. Label Hypotenuse, adjacent and opposite sides
2. Based on given, use sin, cos, or tan function
3. Then using your answer from #2 use the reverse function on your answer to get your final angle measurement.

Question 21

What is the Complementary Trig Ratios Rule?

Answer 21

If B is a right angle and A and C are complementary angles than:
1. sin(θ) = cos(90° - θ)
2. cos(θ) = sin(90° - θ)

Question 22

What do you always have to remember about sine and cosine’s relationship?

Answer 22

• The sine of an angle is the cosine of its compliment
• The cosine of an angle is the sine of its compliment

Question 23

What do you use when you are given one side and one angle to solve for a mystery side or angle?

Answer 23

Use Trigonometric Ratios (Yuck)

Question 24

When given a missing side with two given sides in a right triangle use…

Answer 24

The Pythagorean Theorem!!!!!! (YayYY)

Question 25

When given two sides and a mystery angle use

Answer 25

Inverse Trigonometric Ratios
👁👄👁

Question 26

When given the sine and cosine of a given angle and asked for the other one use…

Answer 26

Complementary Ratios Strategy

Question 27

What is the formula to find the area of right triangles?

Answer 27

A = 1/2bh

Question 28

What is the formula to find out the area in any other triangles? (not right)

Answer 28

A = 1/2absin(C)

Question 29

What is the Law of Cosines?

Answer 29

c2 = a2 + b2 - 2abcos(C) (Ewwww)

Question 30

What is the Law of Sines?

Answer 30

sin(A)/a = sin(B)/b = sin(C)/c
Basically the sine of an angle and the opposite side will be equal to all the other ratios of sin/opposite side.

Question 31

When should you use the Law of Sines?

Answer 31

You should use the Law of Sines when given two sides and no angle, or if given two angles and a side are known.

Question 32

When should you use the Law of Cosines?

Answer 32

Use when
- Given two sides and the angle between them
- Three sides are known