The Trigonometric Functions we will be

looking at

SINE

COSINE

TANGENT

The Trigonometric Functions

SINE

COSINE

TANGENT

SINE

Pronounced “sign”

Pronounced “co-sign”

COSINE

Pronounced “tan-gent”

TANGENT

Pronounced “theta”

Greek Letter

Represents an unknown angle

oppositehypotenuse

LegOppSin

Hyp

adjacent tan

Opp LegAdj Leg

hypotenuseopposite

adjacent

LegAdjCos

Hyp

Finding sin, cos, and tan.

Just writing a ratio.

Find the sine, the cosine, and the tangent of theta.

Give a fraction.sin

opphyp

3537

cosadjhyp

1237

tanoppadj

3512

35

12

37

Shrink yourself down and stand where the angle is.Now, figure out your ratios.

Find the sine, the cosine, and the tangent of thetaFind the sine, the cosine, and the tangent of theta

24.5

23.1

8.2

8.224.5

23.124.5

8.223.1

Shrink yourself down and stand where the angle is.Now, figure out your ratios.

sinopphyp

cosadjhyp

tanoppadj

Using Trig Ratios Using Trig Ratios to Find a Missing to Find a Missing

SIDESIDE

To fi

nd

a

To fi

nd

a

mis

sin

g

mis

sin

g

SID

ES

IDE

1. Draw stick-man at the given angle.

2. Identify the GIVEN sides (Opposite, Adjacent, or Hypotenuse).

3. Figure out which trig ratio to use.

4. Set up the EQUATION.5. Solve for the variable.

1.1.Problems match the Problems match the WS.WS.

HH

AAcoscos15

9x

9 cos15 x 8.7 x

Where does Where does x reside?x reside?

If you see If you see it up high it up high then we then we

MULTIPMULTIPLY!LY!

2.2.Problems match the Problems match the WS.WS.

HH OO

sin9

sin50x

9sin50

x 11.7x

Where does x reside?Where does x reside?

If X is down If X is down below,below,

The X and The X and the angle the angle

will switch…will switch…

SLIDE & SLIDE & DIVIDEDIVIDE

3.3.Problems match the Problems match the WS.WS.

HH

AAcos

10cos51

x

10cos51

x 15.9x

Steps to finding the missing angle of a right triangle using trigonometric ratios:

1. Redraw the figure and mark on it HYP, OPP, ADJ relative to the unknown angle

5.92 kmHYP

OPP

ADJ

2.6

7

km

Steps to finding the missing angle of a right triangle using trigonometric ratios:

2. For the unknown angle choose the correct trig ratio which can be used to set up an equation

3. Set up the equation

5.92 kmHYP

OPP

ADJ

2.6

7

km

Steps to finding the missing angle of a right triangle using trigonometric ratios:

4. Solve the equation to find the unknown using the inverse of trigonometric ratio.

5.92 kmHYP

OPP

ADJ

2.6

7

km

Your turn

Practice Together:

Find, to one decimal place, the unknown angle in the triangle.

3.1 km 2.1

km

YOU DO:

Find, to 1 decimal place, the unknown angle in the given triangle.

7 m

4 m

Sin-CosineSin-CosineCofunctionCofunction

The Sin-Cosine Cofunction

sin cos(90 )

cos sin(90 )

1. What is sin A?

3034

1517

2. What is Cos C?

3. What is Sin Z?

2426

1213

4. What is Cos X?

5. Sin 28 = ?

cos62

6. Cos 10 = ?

sin80

7. ABC where B = 90.Cos A = 3/5

What is Sin C? 35

8. Sin = Cos 15

What is ?75

Trig Trig Application Application ProblemsProblems

MM2G2c: Solve application problems using the trigonometric ratios.

Depression and ElevationDepression and Elevation

horizontalhorizontal

line of sight

horizontalhorizontalangle of elevation

angle of depression

9. Classify each angle as angle of elevation or angle of depression.

Angle of Depression

Angle of Elevation

Angle of Depression

Angle of Elevation

Example 10

Over 2 miles (horizontal), a road rises 300 feet (vertical). What is the angle of elevation to the nearest degree? 5280 feet – 1 mile

300tan

10,560

2

Example 11

The angle of depression from the top of a tower to a boulder on the ground is 38º. If the tower is 25m high, how far from the base of the tower is the boulder? Round to the nearest whole number.

32x meters

25tan38

x

Example 12

Find the angle of elevation to the top of a tree for an observer who is 31.4 meters from the tree if the observer’s eye is 1.8 meters above the ground and the tree is 23.2 meters tall. Round to the nearest degree.

34

21.4tan

31.4

Example 13

A 75 foot building casts an 82 foot shadow. What is the angle that the sun hits the building? Round to the nearest degree.

48

82tan

75

Example 14

A boat is sailing and spots a shipwreck 650 feet below the water. A diver jumps from the boat and swims 935 feet to reach the wreck. What is the angle of depression from the boat to the shipwreck, to the nearest degree?

44 650

n935

si

Example 15

A 5ft tall bird watcher is standing 50 feet from the base of a large tree. The person measures the angle of elevation to a bird on top of the tree as 71.5°. How tall is the tree? Round to the tenth.

154.4x feet

tan71.550x

Example 16

A block slides down a 45 slope for a total of 2.8 meters. What is the change in the height of the block? Round to the nearest tenth.

2meters

n452.8x

si

Example 17

A projectile has an initial horizontal velocity of 5 meters/second and an initial vertical velocity of 3 meters/second upward. At what angle was the projectile fired, to the nearest degree?

31

3tan

5

Example 18

A construction worker leans his ladder against a building making a 60o angle with the ground. If his ladder is 20 feet long, how far away is the base of the ladder from the building? Round to the nearest tenth.

10x feetcos6060x