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A A The sides of a right -angled triangle are given special names: The hypotenuse, the opposite and the adjacent. The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides refer to another angle (given to us), other than the 90 o.
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Learning Objective:
To be able to describe the sides of right-angled triangle for use in trigonometry.
•Setting up ratios•Trig in the Calculator
Trigonometry is concerned with the connection between the sides and angles in any right angled triangle.
Angle
A
A
The sides of a right -angled triangle are given special names:The hypotenuse, the opposite and the adjacent.The hypotenuse is the longest side and is always opposite the right angle.The opposite and adjacent sides refer to another angle (given to us), other than the 90o.
There are three formulae involved in trigonometry:
sin A=
cos A=
tan A =
S O H C A H T O A
Finding the ratios
The simplest form of question is finding the decimal value of the ratio of a given angle.
Find:
1) sin 32 =sin 32 =
2) cos 23 =
3) tan 78 =
4) tan 27 =
5) sin 68 =
Using ratios to find angles
It can also be used in reverse, finding an angle from a ratio.To do this we use the sin-1, cos-1 and tan-1 function keys. (hitting the 2nd key first)
Example:1. sin x = 0.1115 find angle x.
x = sin-1 (0.1115)x = 6.4o
2. cos x = 0.8988 find angle x
x = cos-1 (0.8988)x = 26o
sin-1 0.1115 =shift sin( )
cos-1 0.8988 =shift cos( )
Ex. 1: Finding Trig Ratios• Compare the
sine, the cosine, and the tangent ratios for A in each triangle beside.
15
817
A
B
C
7.5
48.5
A
B
C
Ex. 1: Finding Trig Ratios
15
817
A
B
C
7.5
48.5
A
B
C
Large Small
sin A =
opposite hypotenuse
cosA =
adjacent hypotenuse
tanA =
opposite adjacent
817
≈ 0.4706
1517
≈ 0.8824
815
≈ 0.5333
48.5
≈ 0.4706
7.58.5
≈ 0.8824
47.5
≈ 0.5333
Trig ratios are often expressed as decimal approximations.
Ex. 2: Finding Trig Ratios
S
sin S =
opposite hypotenuse
cosS =
adjacent hypotenuse
tanS =
opposite adjacent
513
≈ 0.3846
1213
≈ 0.9231
512
≈ 0.4167
adjacent
opposite
12
13 hypotenuse5
R
T S
Ex. 2: Finding Trig Ratios—Find the sine, the cosine, and the tangent of the indicated angle.
R
sin S =
opposite hypotenuse
cosS =
adjacent hypotenuse
tanS =
opposite adjacent
1213
≈ 0.9231
513
≈ 0.3846
125
≈ 2.4
adjacent
opposite12
13 hypotenuse5
R
T S
Examples of Trig Ratios
Sin P
Cos P
1220
16Q
P
Tan P Tan Q
Cos Q
Sin Q1620
1220
1612
1220
1620
1216
Opposite
Similar Triangles and Trig Ratios
ABC QPR
35
4A
B12
20
16Q
P
RC
Tan Q
Cos Q
Sin Q1220
1620
1216
Tan A
Cos A
Sin A35
45
34
