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Trigonometric Ratios. The Trigonometric Functions we will be looking at. SINE. COSINE. TANGENT. The Trigonometric Functions. SIN E. COS INE. TAN GENT. SIN E. Prounounced “sign”. COS INE. Prounounced “co-sign”. TAN GENT. Prounounced “tan-gent”. hypotenuse. hypotenuse. opposite. - PowerPoint PPT Presentation
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TRIGONOMETRIC RATIOS
The Trigonometric Functions we will be
looking at
SINECOSINE
TANGENT
The Trigonometric Functions
SINECOSINE
TANGENT
SINE
Prounounced “sign”
Prounounced “co-sign”
COSINE
Prounounced “tan-gent”
TANGENT
oppositehypotenuse
Sin OppHyp
adjacent
Cos AdjHyp
Tan OppAdj
hypotenuse opposite
adjacent
A
WE CAN USE TRIGONOMETRIC RATIOS TO FIND MISSING SIDESSine= sinA =
Cosine = cosA =
Tangent = tanA =
WHERE A IS THE ANGLE MEASUREMENT
A
B C
THE RATIOS CAN BE REMEMBERED BY THE SAYING
SOH CAH TOAsinA = cosA = tanA =
opposite
adjacent
hypotenuse
A
What is the sine ratio of A?
What is the cosine ratio of A?
What is the tangent ratio of A?
A
B C
108
6
Find the sine, the cosine, and the tangent of angle A.Give a fraction and decimal answer (round to 4 places).
hypooppAsin
8.109
8333.
hypoadjAcos
8.106
5555.
adjoppAtan
69
5.1
9
6
10.8
A
Find the values of the three trigonometric functions of .
4
3
? Pythagorean Theorem:(3)² + (4)² = c²
5 = c
opphyp
45
adjhyp
35
oppadj
43
sin cos tan
5
Find the sine, the cosine, and the tangent of angle A
A
24.5
23.1
8.2
hypooppAsin 5.24
2.8 3347.
hypoadjAcos
5.241.23
9429.
adjoppAtan
1.232.8
3550.
B Give a fraction and decimal answer (round to 4 decimal places).
HOW CAN WE USE THESE RATIOS TO FIND SIDE LENGTHS?
What is the length of BC?A
B C
16
22ºAsk yourself the following questions~What angle measurement do I have?A~From that angle what am I LOOKING FOR (adjacent, opposite or hypotenuse)OPPOSITE~From the angle what do I HAVE? (adjacent, opposite or hypotenuse)HYPOTENUSE~So which ratio should we use?SINE
HOW CAN WE USE THESE RATIOS TO FIND SIDE LENGTHS?
What is the length of BC?A
B C
16
22ºSin22 =
16Sin22 =x
5.99=x
D
E F14
65º
Find the length of ED. Which ratioshould we use? Solve.
Find the length of DF. Which ratioShould we use? Solve.
FINDING AN ANGLE.(FIGURING OUT WHICH RATIO TO USE AND GETTING TO USE THE 2ND BUTTON AND ONE OF THE TRIG BUTTONS.)
EX. 1: FIND . ROUND TO FOUR DECIMAL PLACES.
9
17.2
Make sure you are in degree mode (not radian).
92.17tan
2nd tan 17.2 9
3789.62
EX. 2: FIND . ROUND TO THREE DECIMAL PLACES.
23
7
Make sure you are in degree mode (not radian).
237cos
2nd cos
7 23
281.72
EX. 3: FIND . ROUND TO THREE DECIMAL PLACES.
400
200
Make sure you are in degree mode (not radian).
400200sin
2nd sin 200 400
30
WHEN WE ARE TRYING TO FIND A SIDEWE USE SIN, COS, OR TAN.
When we are trying to find an angle we use sin-1, cos-1, or tan-1.
SOLVE THE PROBLEM:
A ladder is leaning against a house so that the top of the ladder is 12 feet above the ground. The angle with the ground is 51º. How far is the base of the ladder from the house?
12
51º
x
An angle of elevation is the angle formed by a horizontal line and a line of sight to a point above the line. In the diagram, 1 is the angle of elevation from the tower T to the plane P.An angle of depression is the angle formed by a horizontal line and a line of sight to a point below the line. 2 is the angle of depression from the plane to the tower.
Since horizontal lines are parallel, 1 2 by the Alternate Interior Angles Theorem. Therefore the angle of elevation from one point is congruentto the angle of depression from the other point.
The sun hits the top of a tree which creates a shadow 15 feet long. If the angle of elevation from the ground to the top of the tree is 42º, how tall is the tree?
tan42=
x= 15tan42
x= 13.51 ft
42º
x
15
42º
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?
200x60°
cos 60°
x (cos 60°) = 200
x
X = 400 yards
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° 50
y
y = 50 (tan 71.5°)
y = 50 (2.98868) 149.4y ft
Ex.
OppHyp
A plane is flying at an altitude of 1.5km. The pilot wants to descend into an airport so that the path of the plane makes an angle of 5° with the ground. How far from the airport (horizontal distance) should the descent begin?
HAPPY LANDING
1.5km5°
x
RIVER WIDTHA surveyor is measuring a river’s width. He uses a tree and a big rock that are on the edge of the river on opposite sides. After turning through an angle of 90° at the big rock, he walks 100 meters away to his tent. He finds the angle from his walking path to the tree on the opposite side to be 25°. What is the width of the river?Draw a diagram to describe this situation. Label the variable(s)
RIVER WIDTH
tan(25 )100d
We are looking at the “opposite” and the “adjacent” from the given angle, so we will use tangent
Multiply by 100 on both sides
100 tan(25 ) d
46.63 metersd
BUILDING HEIGHTA spire sits on top of the top floor of a building. From a point 500 ft. from the base of a building, the angle of elevation to the top floor of the building is 35o. The angle of elevation to the top of the spire is 38o. How tall is the spire?
Construct the required triangles and label.
500 ft.
38o 35o
BUILDING HEIGHTWrite an equation and solve.
Total height (t) = building height (b) + spire height (s)
500 ft.
38o 35o
Solve for the spire height.
t
b
s
Total Height
tan(38 )500
o t
500 tan(38 )o t
BUILDING HEIGHTWrite an equation and solve.
500 ft.
38o 35o
Building Height
tan(35 )500
o b
500 tan(35 )o b t
b
s
BUILDING HEIGHT
5050 0 t0 tan an(3(38 ) 5 )o o s
Write an equation and solve.
500 ft.
38o 35o
500 tan(38 )o t 500 tan(35 )o b
5050 0 t0 tan an(3(38 ) 5 )o o s
The height of the spire is approximately 41 feet.
t
b
s
Total height (t) = building height (b) + spire height (s)