Have you ever thought about how we can find out how tall a skyscraper is without actually climbing it with a tape measure? We employ trigonometry ratios. These ratios show how the sides and angles of a triangle are related to each other mathematically. For children, learning these is like acquiring a "universal ruler" that can measure anything from the distance between stars to the height of a mountain.
What are Trigonometric Ratios?
The trigonometric ratios definition is the relationship between the measurement of the angles and the length of the sides of a right-angled triangle.
There are three sides in every right-angled triangle that are related to an angle (called $\theta$):
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Hypotenuse: The longest side is the hypotenuse, which is opposite the 90° angle.
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Opposite: The side that is directly across from the angle $\theta$.
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Adjacent: The side that is next to angle $\theta$.
Trigonometric Ratios Formula
The SOH CAH TOA formula is the most crucial thing for pupils to learn about trigonometric ratios. This is a magic word that will help you recall the first three ratios:
|
Ratio
|
Name
|
Formula
|
Memory Aid
|
|
$\sin \theta$
|
Sine
|
$\frac{\text{Opposite}}{\text{Hypotenuse}}$
|
Sine = Opposite / Hypotenuse
|
|
$\cos \theta$
|
Cosine
|
$\frac{\text{Adjacent}}{\text{Hypotenuse}}$
|
Cosine = Adjacent / Hypotenuse
|
|
$\tan \theta$
|
Tangent
|
$\frac{\text{Opposite}}{\text{Adjacent}}$
|
Tangent = Opposite / Adjacent
|
The Reciprocal Ratios
There are three more ratios that are just the "flipped" versions of the ones above:
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$\csc \theta$ (Cosecant): $\frac{\text{Hypotenuse}}{\text{Opposite}}$ (Inverse of Sine)
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$\sec \theta$ (Secant): $\frac{\text{Hypotenuse}}{\text{Adjacent}}$ (Inverse of Cosine)
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$\cot \theta$ (Cotangent): $\frac{\text{Adjacent}}{\text{Opposite}}$ (Inverse of Tangent)
Trigonometric Ratios Table for Standard Angles
A quick values table saves time in exams because these angles show up everywhere. Once you know the patterns for 0°, 30°, 45°, 60°, 90°, you can also work out many other values by using identities (like complementary angles) and by spotting symmetry.
|
Angle
|
sinθ
|
cosθ
|
tanθ
|
cosecθ
|
secθ
|
cotθ
|
|
0°
|
0
|
1
|
0
|
Not Defined
|
1
|
Not Defined
|
|
30°
|
1/2
|
√3/2
|
1/√3
|
2
|
2/√3
|
√3
|
|
45°
|
1/√2
|
1/√2
|
1
|
√2
|
√2
|
1
|
|
60°
|
√3/2
|
1/2
|
√3
|
2/√3
|
2
|
1/√3
|
|
90°
|
1
|
0
|
Not Defined
|
1
|
Not Defined
|
0
|
Read More - Average Formula in Maths with Easy Examples
Trigonometric Ratio Identities
Identities help you simplify expressions, verify answers, and compute values without a calculator.
1) Complementary Angle Identities (90° − θ)
sin(90° − θ) = cosθ
cos(90° − θ) = sinθ
tan(90° − θ) = cotθ
cot(90° − θ) = tanθ
sec(90° − θ) = cosecθ
cosec(90° − θ) = secθ
2) Pythagorean Identities
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
3) Sum and Difference Formulas
sin(A ± B) = sinA cosB ± cosA sinB
cos(A ± B) = cosA cosB ∓ sinA sinB
tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)
4) Product-to-Sum Handy Forms
2sinA cosB = sin(A + B) + sin(A − B)
2cosA cosB = cos(A + B) + cos(A − B)
2sinA sinB = cos(A − B) − cos(A + B)
5) Double, Half, and Triple Angle Highlights
sin2θ = 2sinθ cosθ
cos2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
tan2θ = (2tanθ)/(1 − tan²θ)
sin(θ/2) = ±√[(1 − cosθ)/2], cos(θ/2) = ±√[(1 + cosθ)/2]
sin3θ = 3sinθ − 4sin³θ, cos3θ = 4cos³θ − 3cosθ
Read More - Types of Functions in Maths – Graphs & Examples
Trigonometric Ratios Examples
Let's see what are trigonometric ratios in a maths situation from real world.
The Problem: You have a triangle with an Opposite side of 3 cm, an Adjacent side of 4 cm, and a Hypotenuse of 5 cm. Find the Sine, Cosine, and Tangent.
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Step 1 (Sine): $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3}{5} = \mathbf{0.6}$
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Step 2 (Cosine): $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{5} = \mathbf{0.8}$
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Step 3 (Tangent): $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{3}{4} = \mathbf{0.75}$
Why are Trigonometric Ratios Important for students?
Knowing how to use the trigonometric ratios formula can help you get a lot of amazing jobs:
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Video Games: Mario uses these ratios to figure out how to leap over a gap!
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Architecture: To build a stable roof or a staircase, architects must know these ratios.
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Navigation: Ship captains use them to find their way across the ocean.
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