Click here for the previous basics of trigonometry.
Now we will study the next part of trigonometry.
Today we will study trigonometric ratios of 0°, 30°, 45°, and 60°.
Here angle AOP is 0°.
According to the diagram, OP = r, and the coordinates of point P are (r, 0). So the x-coordinate of point P is r and the y-coordinate of point P is 0.
So, all the trigonometric ratios of q = 0° with
x = r,
y = 0,
r = r.
a) sin 0° = y/r
sin 0° = 0/r
sin 0° = 0
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b) cos 0° = x/r
cos 0° = r/r
cos 0° = 1
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c) tan 0° = y/x
tan 0° = 0/r
tan 0° = 0
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d) csc 0° = r/y
csc 0° = r/0
csc 0° = ∞
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e) sec 0° = r/x
sec 0° = r/r
sec 0° = 1
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f) cot 0° = x/y
cot 0° = r/0
cot 0° = ∞
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2) Trigonometric ratios of an angle q = 30°
Here angle AOP is 30°.
We know that the side opposite to 30° is half the hypotenuse and the side opposite to 60° is √3/ 2 times the hypotenuse.
So if hypotenuse OP = r, then AP = r/2 and OA = (√3 r)/ 2.
Here the coordinates of point P will be P ((√3 r)/ 2, r/2).
Here the coordinates of point P will be P ((√3 r)/ 2, r/2).
So, all the trigonometric ratios of q = 30° with
x = (√3 r)/ 2,
y = r/2,
r = r.
a) sin 30° = y/r
sin 30° = (r/2)/r
sin 30° = 1/2
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b) cos 30° = x/r
cos 30° = [(√3 r)/2]/r cos 30° = √3/2
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c) tan 30° = y/x
tan 30° = (r/2)/(√3 r)/2
tan 30° = 1/√3
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d) csc 30° = r/y
csc 30° = r/(r/2)
csc 30° = 2
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e) sec 30° = r/x
sec 30° = r/[(√3 r)/2]
sec 30° = 2/√3
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f) cot 30° = x/y
cot 30° = [(√3 r)/ 2]/(r/2)
cot 30° = √3
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We know that the side opposite 45° is 1/√2 times the hypotenuse.
So if hypotenuse OP = r, then AP = r/√2 and OA = r/√2.
Here the coordinates of point P will be P (r/√2, r/√2).
Here the coordinates of point P will be P (r/√2, r/√2).
So, all the trigonometric ratios of q = 45° with
x = r/√2,
y = r/√2,
r = r.
a) sin 45° = y/r
sin 45° = [r/√2]/r
sin 45° = 1/√2
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b) cos 45° = x/r
cos 45° = [r/√2]/r
cos 45° = 1/√2
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c) tan 45° = y/x
tan 45° = [r/√2]/ [r/√2]
tan 45° = 1
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d) csc 45° = r/y
csc 45° = r/[r/√2]
csc 45° = √2
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e) sec 45° = r/x
sec 45° = r/[r/√2]
sec 45° = √2
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f) cot 45° = x/y
cot 45° = [r/√2]/ [r/√2]
cot 45° = 1
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We know that the side opposite 60° is √3/2 times the hypotenuse and the side opposite 30° is half times the hypotenuse.
So if hypotenuse OP = r, then AP = (√3 r)/2 and OA = r/2.
Here the coordinates of point P will be P (r/2, (√3 r)/2).
Here the coordinates of point P will be P (r/2, (√3 r)/2).
So, all the trigonometric ratios of q = 60° with
x = r/2,
y = (√3 r)/2,
r = r.
a) sin 60° = y/r
sin 60° = [(√3 r)/2]/r
sin 60° = √3/2
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b) cos 60° = x/r
cos 60° = [r/2]/r
cos 60° = 1/2 |
c) tan 60° = y/x
tan 60° = [(√3 r)/2]/[r/2]
tan 60° = √3
|
d) csc 60° = r/y
csc 60° = r/[(√3 r)/2]
csc 60° = 2/√3 |
e) sec 60° = r/x
sec 60° = r/[r/2]
sec 60° = 2
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f) cot 60° = x/y
cot 60° = [r/2]/[(√3 r)/2]
cot 60° = 1/√3
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In the next blog, we will study the trigonometric ratios of 90°, and 180° and the tabulated form of all the trigonometric ratios.
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