Trigonometric

RatiosA RATIO is a comparison of two numbers. For example;

boys to girls cats : dogs

right : wrong.

In Trigonometry, the comparison is between sides of a triangle ( right triangle).

CCSS: G.SRT.7

EXPLAIN and USE the relationship between the sine and cosine of

complementary angles.

Standards for Mathematical Practice

• 1. Make sense of problems and persevere in solving them.

• 2. Reason abstractly and quantitatively. • 3. Construct viable arguments and critique the

reasoning of others.  • 4. Model with mathematics. • 5. Use appropriate tools strategically. • 6. Attend to precision. • 7. Look for and make use of structure. • 8. Look for and express regularity in repeated reasoning.

Warm up

• Solve the equations:

• A) 0.875 = x/18

• B) 24/y = .5

• C) y/25 = .96

E.Q:

How can we find the sin, cosine, and the tangent of an acute angle?How do we use trigonometric ratios to solve real-life problems?

Trig. Ratios

Name“say”

Sine Cosine tangent

AbbreviationAbbrev.

Sin Cos Tan

Ratio of an angle measure

Sinθ = opposite side hypotenuse

cosθ = adjacent side hypotenuse

tanθ =opposite side adjacent side

Easy way to remember trig ratios:

SOH CAH TOA

Three Trigonometric Ratios• Sine – abbreviated ‘sin’.

– Ratio: sin θ = opposite side

hypotenuse

• Cosine - abbreviated ‘cos’. – Ratio: cos θ = adjacent side

hypotenuse

• Tangent - abbreviated ‘tan’. – Ratio: tan θ = opposite side

adjacent side

Θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’.

Let’s practice…

B c a

C b A

Write the ratio for sin A

Sin A = o = a h c

Write the ratio for cos A

Cos A = a = b h c

Write the ratio for tan A

Tan A = o = a a b

Let’s switch angles: Find the sin, cos and tan for Angle B:

Sin B = b

cCos B = a

c

Tan B = b

a

Make sure you have a calculator…I want to find Use these calculator keys

sin, cos or tan

ratio

SINCOSTAN

Angle measureSIN-1

COS-1

TAN-1

Set your calculator to ‘Degree’…..

MODE (next to 2nd button)

Degree (third line down… highlight it)

2nd

Quit

Let’s practice…

C2cm

B 3cm A

Find an angle that has a tangent (ratio) of 2

3

Round your answer to the nearest degree.

Process:

I want to find an ANGLE

I was given the sides (ratio)

Tangent is opp

adj

TAN-1(2/3) = 34°

Practice some more…

Find tan A: 24.19 12

A 21

Tan A = opp/adj = 12/21

Tan A = .5714

8

4A

Tan A = 8/4 = 2 8

Find tan A:

Trigonometric Ratios

• When do we use them?– On right triangles that are NOT 45-45-90 or

30-60-90

Find: tan 45

1

Why?

tan = opphyp

Using trig ratios in equations

Remember back in 1st grade when you had to solve:

12 = x What did you do? 6

(6) (6)

72 = xRemember back in 3rd grade when x was in

the denominator? 12 = 6 What did you do? x

(x) (x)

12x = 6__ __12 12 x = 1/2

x cm

15 cm

34°

Ask yourself:In relation to the angle,

what pieces do I have?Opposite and hypotenuse

Ask yourself:

What trig ratio uses Opposite and Hypotenuse?

SINE

Set up the equation and solve:

Sin 34 = x 15

(15) (15)

(15)Sin 34 = x8.39 cm = x

x cm

12 cm53°

Ask yourself:In relation to the angle,

what pieces do I have?Opposite and adjacent

Ask yourself:

What trig ratio uses Opposite and adjacent?

tangent

Set up the equation and solve:

Tan 53 = x 12

(12) (12)

(12)tan 53 = x15.92 cm = x

x cm

18 cm

68°

Ask yourself:In relation to the angle,

what pieces do I have?Adjacent and hypotenuse

Ask yourself:

What trig ratio uses adjacent and hypotnuse?

cosine

Set up the equation and solve:Cos 68 = 18 x

(x) (x)

(x)Cos 68 = 18

X = 18 cos 68

_____ _____cos 68 cos 68

X = 48.05 cm

42 cm22 cm

θ

This time, you’re looking for theta. Ask yourself:In relation to the angle, what pieces do I have? Opposite and hypotenuse

Ask yourself:

What trig ratio uses opposite and hypotenuse? sine

Set up the equation (remember you’re looking for theta):Sin θ = 22 42

Remember to use the inverse function when you find theta

THIS IS IMPORTANT!!

Sin -1 22 = θ 42

31.59°= θ

17 cm

22 cm

θ

You’re still looking for theta.

Ask yourself:

What trig ratio uses the parts I was given? tangent

Set it up, solve it, tell me what you get.

tan θ = 17 22

THIS IS IMPORTANT!!

tan -1 17 = θ 22

37.69°= θ

Using trig ratios in equations

Remember back in 1st grade when you had to solve:

12 = x What did you do? 6

(6) (6)

72 = xRemember back in 3rd grade when x was in

the denominator? 12 = 6 What did you do? x

(x) (x)

12x = 6__ __12 12 x = 1/2

Types of Angles

• The angle that your line of sight makes with a line drawn horizontally.

• Angle of Elevation

• Angle of Depression

Horizontal Line

Angle of Elevation

Line of Sight

Horizontal Line

Line of Sight

Angle of Depression

Indirect Measurement

45 ft59

h

tan 59 = opposite

adjacent

tan 59 = h

45

45 tan 59 = h

74.9 h

SOA CAH TOA

SOA CAH TOA

Solving a right triangle

• Every right triangle has one right angle, two acute angles, one hypotenuse and two legs. To solve a right triangle, means to determine the measures of all six (6) parts. You can solve a right triangle if the following one of the two situations exist:– Two side lengths– One side length and one acute angle

measure

E.Q• How do we use right triangles to solve

real life problems?

Note:

• As you learned in Lesson 9.5, you can use the side lengths of a right triangle to find trigonometric ratios for the acute angles of the triangle. As you will see in this lesson, once you know the sine, cosine, or tangent of an acute angle, you can use a calculator to find the measure of the angle.

WRITE THIS DOWN!!!• In general, for an acute angle A:

– If sin A = x, then sin-1 x = mA– If cos A = y, then cos-1 y = mA– If tan A = z, then tan-1 z = mA

The expression sin-1 x is read as “the inverse sine of x.”

• On your calculator, this means you will be punching the 2nd function button usually in yellow prior to doing the calculation. This is to find the degree of the angle.

Example 1:

• Solve the right triangle. Round the decimals to the nearest tenth.

c

3 2

B

C

A

HINT: Start by using the Pythagorean Theorem. You have side a and side b. You don’t have the hypotenuse which is side c—directly across from the right angle.

Example 1:

c

3 2

B

C

A

(hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem

c2 = 32 + 22 Substitute values

c2 = 9 + 4 SimplifySimplify

c = √13c2 = 13

Find the positive square rootc ≈ 3.6 Use a calculator to approximate

Example 1 continued

• Then use a calculator to find the measure of B:

2nd functionTangent button2Divided by symbol3 ≈ 33.7°

Finally

• Because A and B are complements, you can write

mA = 90° - mB ≈ 90° - 33.7° = 56.3°

The side lengths of the triangle are 2, 3 and √13, or about 3.6. The triangle has one right angle and two acute angles whose measure are about 33.7° and 56.3°.

Ex. 2: Solving a Right Triangle (h)

• Solve the right triangle. Round decimals to the nearest tenth.

13 h

gH J

G

25°You are looking for opposite and hypotenuse which is the sin ratio.

sin H =opp.

hyp.

13 sin 25° =h

1313

13(0.4226) ≈ h

5.5 ≈ h

Set up the correct ratio

Substitute values/multiply by reciprocal

Substitute value from table or calculator

Use your calculator to approximate.

Ex. 2: Solving a Right Triangle (g)

• Solve the right triangle. Round decimals to the nearest tenth.

13 h

gH J

G

25°

You are looking for adjacent and hypotenuse which is the cosine ratio.

cos G =adj.

hyp.

13 cos 25° =g

1313

13(0.9063) ≈ g

11.8 ≈ h

Set up the correct ratio

Substitute values/multiply by reciprocal

Substitute value from table or calculator

Use your calculator to approximate.

Using Right Triangles in Real Life

• Space Shuttle: During its approach to Earth, the space shuttle’s glide angle changes.

• A. When the shuttle’s altitude is about 15.7 miles, its horizontal distance to the runway is about 59 miles. What is its glide angle? Round your answer to the nearest tenth.

Solution:

• You know opposite and adjacent sides. If you take the opposite and divide it by the adjacent sides, then take the inverse tangent of the ratio, this will yield you the slide angle.

distance to runway

altitudeGlide = x°

59 miles

15.7 miles

tan x° =opp.

adj.

tan x° =15.7

59

Key in calculator 2nd function,

tan 15.7/59 ≈ 14.9

Use correct ratio

Substitute values

When the space shuttle’s altitude is about 15.7 miles, the glide angle is about 14.9°.

B. Solution• When the space

shuttle is 5 miles from the runway, its glide angle is about 19°. Find the shuttle’s altitude at this point in its descent. Round your answer to the nearest tenth.

distance to runway

altitudeGlide = 19°

5 miles

h

tan 19° =opp.

adj.

tan 19° =h

5

Use correct ratio

Substitute values

5 tan 19° =h

5

5 Isolate h by multiplying by 5.

1.7 ≈ h Approximate using calculator The shuttle’s altitude is about 1.7 miles.