Right Triangle Trigonometry

Objectives

• Find trigonometric ratios using right triangles.

• Use trigonometric ratios to find angle measures in right triangles.

History Right triangle trigonometry is the study of the relationship between the sides and angles of right triangles. These relationships can be usedto make indirect measurements like those using similar triangles.

Trigonometric Ratios

Only Apply to Right Triangles

The 3 Trigonometric Ratios

• The 3 ratios are Sine, Cosine and TangentOpposite SideSine RatioHypotenuse

sin Adjacent SideCo e RatioHypotenuse

Opposite SideTangent RatioAdjacent Side

Chief SohCahToa

The Amazing Legend of…

The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. The sides of the right triangle are:

the side opposite the acute angle ,

the side adjacent to the acute angle , and the hypotenuse of the right triangle.

The trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant.

opp

adj

hyp

θ

sin 𝜃=𝑜𝑝𝑝h𝑦𝑝 cos𝜃=

𝑎𝑑𝑗h𝑦𝑝 tan𝜃=

𝑜𝑝𝑝𝑎𝑑𝑗

csc 𝜃=h𝑦𝑝𝑜𝑝𝑝 sec𝜃=

h𝑦𝑝𝑎𝑑𝑗 cot 𝜃=

𝑎𝑑𝑗𝑜𝑝𝑝

EVALUATING TRIGONOMETRIC FUNCTIONS𝑎=4 𝑎𝑛𝑑𝑏=3

Find all six trig functions of angle A

Remember SOH CAH TOA and the reciprocal identities

sin 𝜃=𝑜𝑝𝑝h𝑦𝑝

cos𝜃=𝑎𝑑𝑗h𝑦𝑝

tan𝜃=𝑜𝑝𝑝𝑎𝑑𝑗

csc 𝜃=h𝑦𝑝𝑜𝑝𝑝

sec𝜃=h𝑦𝑝𝑎𝑑𝑗

cot 𝜃=𝑎𝑑𝑗𝑜𝑝𝑝

What is the value of h?

¿𝟒

¿𝟑

𝟓45

54

35

53

43

34

EVALUATING TRIGONOMETRIC FUNCTIONS𝑎=12𝑎𝑛𝑑𝑏=5

Find all six trig functions of angle A

Remember SOH CAH TOA and the reciprocal identities

sin 𝜃=𝑜𝑝𝑝h𝑦𝑝

cos𝜃=𝑎𝑑𝑗h𝑦𝑝

tan𝜃=𝑜𝑝𝑝𝑎𝑑𝑗

csc 𝜃=h𝑦𝑝𝑜𝑝𝑝

sec𝜃=h𝑦𝑝𝑎𝑑𝑗

cot 𝜃=𝑎𝑑𝑗𝑜𝑝𝑝

What is the value of h?

¿𝟏𝟐

¿𝟓

𝟏𝟑1213

1312

513

135

125

512

EVALUATING TRIGONOMETRIC FUNCTIONS𝑎=1𝑎𝑛𝑑 h=3

Find all six trig functions of angle A

Remember SOH CAH TOA and the reciprocal identities

sin 𝜃=𝑜𝑝𝑝h𝑦𝑝

cos𝜃=𝑎𝑑𝑗h𝑦𝑝

tan𝜃=𝑜𝑝𝑝𝑎𝑑𝑗

csc 𝜃=h𝑦𝑝𝑜𝑝𝑝

sec𝜃=h𝑦𝑝𝑎𝑑𝑗

cot 𝜃=𝑎𝑑𝑗𝑜𝑝𝑝

What is the value of b?

¿𝟏

¿𝟐√𝟐

𝟑13 3

2√23

3√24

√24

2√2

Calculate the trigonometric functions for a 45 angle.

2

1

1

45

csc 45 = = =

12 2

opphypsec 45 = = =

12 2

adjhyp

cos 45 = = =

22

21

hypadjsin 45 = = =

22

21

hypopp

cot 45 = = = 1

oppadj

11tan 45 = = = 1

adjopp

11

60○ 60○

Consider an equilateral triangle with each side of length 2.

The perpendicular bisector

of the base bisects the opposite angle.

The three sides are equal, so the angles are equal; each is 60.

Geometry of the 30-60-90 triangle

2 2

21 1

30○ 30○

3

Use the Pythagorean Theorem to find the length of the altitude, .

Calculate the trigonometric functions for a 30 angle.

12

303

sin 30 °=𝑜𝑝𝑝h𝑦𝑝 =

12

cos 30 °= 𝑎𝑑𝑗h𝑦𝑝=√3

2

tan 30 °=𝑜𝑝𝑝𝑎𝑑𝑗 = 1

√3=√3

3 cot 30 °= 𝑎𝑑𝑗h𝑦𝑝=√3

1=√3

sec 30°=h𝑦𝑝𝑎𝑑𝑗 =2√3

=2√33

csc 30 °= h𝑦𝑝𝑜𝑝𝑝=21=2

Calculate the trigonometric functions for a 60 angle.

12 60

3

sin 60 °=𝑜𝑝𝑝h𝑦𝑝 =√3

2

cos 60 °= 𝑎𝑑𝑗h𝑦𝑝=

12

tan 60 °=𝑜𝑝𝑝𝑎𝑑𝑗 =√3

1=√3 cot 60 °= 𝑎𝑑𝑗

h𝑦𝑝= 1√3

=√33

sec 60°=h𝑦𝑝𝑎𝑑𝑗 =21=2

csc 60 °= h𝑦𝑝𝑜𝑝𝑝= 2√3

=2√33

TRIG FUNCTIONS & COMPLEMENTSTwo positive angles are complements if the sum of their measures is .

Example: are complement because .

The sum of the measures of the angles in a triangle is . In a right triangle, we have a angle. That means that the sum of the other two angles is . Those two angles are acute and complement.

If the degree measure of one acute angle is , then the degree measure of the other angle is .

TRIG FUNCTIONS & COMPLEMENTSCompare and .

Therefore, . If two angles are complements, the sine of one equals the cosine of the other.

sin 𝜃=cos ( 𝜋2 −𝜃) cos𝜃=sin ( 𝜋2 −𝜃)

tan𝜃=co t (𝜋2 −𝜃) cot 𝜃= tan ( 𝜋2 −𝜃)

sec𝜃=csc (𝜋2 −𝜃) csc 𝜃=𝑠𝑒𝑐 (𝜋2 −𝜃)

Using cofunction identitiesFind a cofunction with the same value as the given expression:

Find a cofunction with the same value as the given expression:

¿ sec ( 𝜋2 −𝜋3 )

An angle formed by a horizontal line and the line of sight to an object that is above the horizontal line is called the angle of elevation. The angle formedby a horizontal line and the line of sight to an objectthat is below the horizontal line is called the angle ofdepression. Transits and sextants are instrumentsused to measure such angles.

Angle of Elevation

Angle of Depression

Angle of ELEVATION AND DEPRESSION

A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree?

50

71.5°

?

tan 71.5°

tan 71.5°50y

y = 50 (tan 71.5°) y = 50 (2.98868) 149.4y ft

OppAdj

Look at the given info. What trig function can we use?

A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge?

200x

60°

cos 60°

x (cos 60°) = 200

x

X = 400 yardsLook at the given information. Which trig function should we use?

570tan 0 x

h = (13.74 + 2) meters

A guy wire from a point 2 m from the top of an electric post makes an angle of 700 with the ground. If the guy wire is anchored 5 m from the base of the post, how high is the pole?

5 m

700

2 m

Guy wire

h = 15.74 meters

x

Which trig function should we use?

Great job, you guys!