2. Trigonometric Ratios
1. Trigonometric Ratios Formed by an Angle
of a Right Triangle
a² + b² = c²
Picture 1. B
A right triangle ABC hypotenuse
β
c a
right angle
side
) α _
A l C
b right angle
side
3. a. Understanding of Sine (sin), Cosine (cos),
and Tangent (tan)
sin α = opposite side / hypotenuse = a / c
cos α = adjacent side / hypotenuse = b / c
tan α = opposite side / adjacent side = a / b
4. Example 1:
Determine the value of sine, cosine and
tangent of BAC and ABC in the triangle
below, if a = 6 and b = 8.
Answers:
From the figure, known that AC = b = 8,
BC = a = 6, and AB = c. The value c can be
calculated by using ‘Pythagoras theorem’.
c² = a² + b²
c² = 6² + 8² = 100
c = √100 = 10
5. then sin BAC = sin α = a/c = 6/10
cos BAC = cos α = b/c = 8/10
tan BAC = tan α = a/b = 6/8
sin ABC = sin β = b/c = 8/10
cos ABC = cos β = a/c = 6/10
tan ABC = tan β = b/a = 8/6
A
α
c
8
_
B ) β 6 l C
6. Note:
In addition has the reverse function the
trigonometry ratios also have conclution.
If α + β = 90°, then sin α = cos β.
Example:
sin 30° = cos 60°,
sin 40° = cos 50°,
sin 20° = cos 70° and so on.
7. b. The Value of Trigonometric Ratios for
Specific Angles
For Angle 45°
Picture 2. Trigonometric ratios of angle 45 °
C
D C
45°
d √2
1 1
A 45° B A 45° B
1 1
8. Table 1. Trigonometric Ratios of Specific
Angle
α 0° 30° 45° 60° 90°
sin α 0 1/2 1/2√2 1/2√3 1
cos α 1 1/2√3 1/2√2 1/2 0
tan α 0 1/3√3 1 √3 ∞
9. 2. Trigonometric Ratios Formula of Related
Angles
a. Trigonometric Ratios in Quadrant I
Y
B’ (x’ , y’)
Picture 3.
x’ y=x Angle (90° - α)
y’
B(x , y)
in Quadrant I
r
α r y
α
X
0 x
10. The value of trigonometry ratios for angle
(90° - α) is the following:
sin (90° - α) = y’/r = x/r = cos α
cos (90° - α) = x’/r = y/r = sin α
tan (90° - α) = y’/x’ = x/y = cot α
csc (90° - α) = r/y’ = r/x = sec α
sec (90° - α) = r/x’ = r/y = csc α
cot (90° - α) = x’/y’ = y/x = tan α
11. b. The Trigonometric Ratios in Quadrant II
Picture 4. The Trigonometric Ratios
in Quadrant II
Y
B’ (x’ , y) B (x , y)
r (180° - α) r
y’ y
x’ x X
α α
0
12. The trigonometric ratios value of angle
(180° - α) is as follow:
sin (180° - α) = y’/r = y/r = sin α
cos (180° - α) = x’/r = - x/r = - cos α
tan (180° - α) = y’/x’ = - y/x = - tanα
csc (180° - α) = r/y’ = r/y = csc α
sec (180° - α) = r/x’ = - r/x = - sec α
cot (180° - α) = x’/y’ = - x/y = - cot α
13. c. Trigonometric Ratios in Quadrant III
Picture 5. Trigonometric Ratios
in Quadrant III
Y B (x , y)
(180° + α)
y
x’ α x
α X
0
y’
B’ (x’ , y)
14. The trigonometric ratios value of angle
(180° + α) is as follow:
sin (180° + α) = y’/r = - y/r = sin α
csc (180° +α) = r/y’ = - r/y = - csc α
cos (180° + α) = x’/r = - x/r = - cos α
sec (180° +α) = r/x’ = - r/x = - sec α
tan (180° + α) = y’/x’ = y/x = tan α
cot (180° + α) = x’/y’ = x/y = cot α
15. d. Trigonometric Ratios in Quadrant IV
Picture 6.
Y B (x , y)
Trigonometric Ratios
r y in Quadran IV
(360° - α)
α
0 α
X
r
B’ (x’ , y’)
16. The trigonometric ratios value of angle
(360° - α) is as follow:
sin (360° - α) = y’/r = - y/r = - sin α
csc (360° -α) = r/y’ = - r/y = - csc α
cos (360° - α) = x’/r = x/r = cos α
sec (360° -α) = r/x’ = r/x = sec α
tan (360° - α) = y’/x’ = - y/x = - tan α
cot (360° - α) = x’/y’ = - x/y = - cot α
17. Trigonometric Identity
There are four trigonometric identities
formula that you should know, as follow:
a. tan α = sin α / cos α , cot α = cos α / sin α
b. sin² α + cos² α = 1
c. 1 + tan² α = sec² α
d. 1 + cot² α = csc² α
