Right Triangle TrigonometryTrigonometry is based upon ratios of the sides of right triangles.
The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle.
θ
opphyp
adjThe 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.
A
A
The hypotenuse is the longest side and is always opposite the right angle.
The opposite and adjacent sides refer to another angle, other than the 90o.
Right Triangle Trigonometry
S O H C A H T O A
The trigonometric functions are: sine, cosine, tangent, cotangent, secant, and cosecant.
opp
adj
hyp
θ
sin = cos = tan =
csc = sec = cot = opp
hypadjhyp
hypadj
adjopp
oppadj
hypopp
Trigonometric Ratios
Finding the ratios
The simplest form of question is finding the decimal value of the ratio of a given angle.
Find using calculator:
1) sin 30 =
sin 30 =
2) cos 23 =
3) tan 78 =
4) tan 27 =
5) sin 68 =
Using ratios to find anglesIt 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.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 =
2nd sin( )
cos-1 0.8988 =
2nd cos( )
Calculate the trigonometric functions for .
The six trig ratios are 4
3
5
sin =54
tan =34
sec =35
cos =53
cot =43
csc =45
cos α =54
sin α =53
cot α =34
tan α =43
csc α =35
sec α =45
What is the relationship of
α and θ?
They are complementary (α = 90 – θ)
Calculate the trigonometric functions for .
Finding an angle from a triangle
To find a missing angle from a right-angled triangle we need to know two of the sides of the triangle.
We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio.
Find angle C
a) Identify/label the names of the sides.
b) Choose the ratio that contains BOTH of the letters.
14 cm
6 cmC
1.
C = cos-1 (0.4286) C = 64.6o
14 cm
6 cmC
1.h
a
We have been given the adjacent and hypotenuse so we use COSINE:
Cos A = hypotenuseadjacent
Cos A = ha
Cos C = 146
Cos C = 0.4286
Find angle x2.
8 cm
3 cmx
a
o
Given adj and oppneed to use tan:
Tan A = adjacentopposite
x = tan-1 (2.6667) x = 69.4o
Tan A = ao
Tan x = 38
Tan x = 2.6667
3.
12 cm10 cmy
Given opp and hypneed to use sin:
Sin A = hypotenuseopposite
x = sin-1 (0.8333) x = 56.4o
sin A = ho
sin x = 1210
sin x = 0.8333
Cos 30 x 7 = k 6.1 cm = k
7 cm
k30o
4. We have been given the adj and hyp so we use COSINE:
Cos A = hypotenuseadjacent
Cos A = ha
Cos 30 = 7k
Finding a side from a triangle
Tan 50 x 4 = r
4.8 cm = r
4 cm
r
50o
5.
Tan A = ao
Tan 50 = 4r
We have been given the opp and adj so we use TAN:
Tan A =
Sin 25 x 12 = k
5.1 cm = k
12 cm k
25o
6.
sin A = ho
sin 25 = 12k
We have been given the opp and hyp so we use SINE:
Sin A =
x =
x
5 cm30o
1. Cos A = ha
Cos 30 =x5
30 cos5
x = 5.8 cm
4 cm
r
50o
2.
Tan 50 x 4 = r
4.8 cm = r
Tan A = aoTan 50 = 4
r
3.
12 cm10 cm
y
y = sin-1 (0.8333) y = 56.4o
sin A = ho
sin y = 1210
sin y = 0.8333
Example: Given sec = 4, find the values of the other five trigonometric functions of .
Solution:
Use the Pythagorean Theorem to solve for the third side of the triangle.
tan = = cot =115
151
15
sin = csc = =415
154
sin1
cos = sec = = 4 41
cos1
15
θ
4
1
Draw a right triangle with an angle such that 4 = sec = = .
adjhyp
14
A surveyor is standing 115 feet from the base of the Washington Monument. The surveyor measures the angle of elevation to the top of the monument as 78.3. How tall is the Washington Monument?
Applications Involving Right Triangles
Solution:
where x = 115 and y is the height of the monument. So, the height of the Washington Monument is
y = x tan 78.3
115(4.82882) 555 feet.
Fundamental Trigonometric Identities
Co function Identitiessin = cos(90 ) cos = sin(90 )sin = cos (π/2 ) cos = sin (π/2 )tan = cot(90 ) cot = tan(90 )tan = cot (π/2 ) cot = tan (π/2 )sec = csc(90 ) csc = sec(90 ) sec = csc (π/2 ) csc = sec (π/2 )
Reciprocal Identities
sin = 1/csc cos = 1/sec tan = 1/cot cot = 1/tan sec = 1/cos csc = 1/sin
Quotient Identities tan = sin /cos cot = cos /sin
Pythagorean Identities
sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2