Trigonometric Ratios in Radians - Values & Examples

Trigonometric Ratios in Radians - Values & Examples

TL;DR

Trigonometric ratios in radians are the same six ratios — sin, cos, tan, csc, sec, cot — measured with the angle written in radians instead of degrees, where a full circle is 2π radians rather than 360°. This article gives the standard-angle table (0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}), how to convert, why radians exist, six worked examples, and the mistakes that cost marks.

What Are Trigonometric Ratios In Radians?

Trigonometric ratios in radians are the values of sine, cosine, tangent, and their reciprocals when the input angle is expressed in radians. The ratio definitions are unchanged — radians are just the second standard way of naming an angle, and the one used everywhere past school.

The six ratios, with θ in radians:

[ \sin \theta, \quad \cos \theta, \quad \tan \theta = \frac{\sin \theta}{\cos \theta} ]

[ \csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta} ]

The mnemonic SOH-CAH-TOA still works for an acute angle in a right triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. SOH-CAH-TOA is taught in degrees, but the right triangle doesn't know the difference — a ( \frac{\pi}{6} ) angle is the same wedge as a 30° angle. The radian label is just the arc-length way of measuring that same opening.

The Trigonometric Ratios In Radians Table (Standard Angles)

Here is the value of every ratio at the five standard angles. This is the table to know cold — almost every radian problem at this level reduces to one of these five inputs.

Angle (deg) Angle (rad) sin⁡θ cos⁡θ tan⁡θ csc⁡θ sec⁡θ cot⁡θ
0° 0 0 1 0 undefined 1 undefined
30° ( \frac{\pi}{6} ) ( \frac{1}{2} ) ( \frac{\sqrt{3}}{2} ) ( \frac{1}{\sqrt{3}} ) 2 ( \frac{2}{\sqrt{3}} ) ( \sqrt{3} )
45° ( \frac{\pi}{4} ) ( \frac{1}{\sqrt{2}} ) ( \frac{1}{\sqrt{2}} ) 1 ( \sqrt{2} ) ( \sqrt{2} ) 1
60° ( \frac{\pi}{3} ) ( \frac{\sqrt{3}}{2} ) ( \frac{1}{2} ) ( \sqrt{3} ) ( \frac{2}{\sqrt{3}} ) 2 ( \frac{1}{
\sqrt{3}} )
90° ( \frac{\pi}{2} ) 1 0 undefined 1 undefined 0

A quicker way to hold the sine row: write ( 0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1 ) for 0, ( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} ). The cosine row is the same list reversed.

How do you convert degrees to radians? Multiply the degree measure by (\frac{\pi}{180}). To go the other way, multiply radians by (\frac{180}{\pi}). So 30° × (\frac{\pi}{180}) = (\frac{\pi}{6}), and (\frac{\pi}{4}) × (\frac{180}{\pi}) = 45°.

Examples Of Trigonometric Ratios In Radians

Example 1

Find ( \sin \frac{\pi}{6} ).

( \frac{\pi}{6} ) is the radian name for 30°.

From the table, ( \sin 30° = \frac{1}{2} ).

Final answer: ( \sin \frac{\pi}{6} = \frac{1}{2} ).

Example 2

Evaluate ( \cos \frac{\pi}{3} + \sin \frac{\pi}{6} ). The correct route is to evaluate each term on its own:

( \cos \frac{\pi}{3} = \frac{1}{2} ) and ( \sin \frac{\pi}{6} = \frac{1}{2} ).

( \frac{1}{2} + \frac{1}{2} = 1 ).

Final answer: 1.

Example 3

Find ( \tan \frac{\pi}{4} ) using the right-triangle definition, then confirm from the table.

( \tan \frac{\pi}{4} = \frac{\text{opposite}}{\text{adjacent}} = 1 ).

The table agrees: ( \tan \frac{\pi}{4} = 1 ).

Final answer: 1.

Example 4

Evaluate ( \csc \frac{\pi}{3} ).

( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} ), thus ( \csc \frac{\pi}{3} = \frac{1}{\sin \frac{\pi}{3}} = \frac{2}{\sqrt{3}} ).

Final answer: ( \csc \frac{\pi}{3} = \frac{2}{\sqrt{3}} ) (or ( \frac{2\sqrt{3}}{3} ) when rationalised).

Example 5

A wheel turns through ( \frac{\pi}{4} ) radians. What point on the unit circle does it land on?

( \frac{\pi}{4} ) is 45°.

( \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}, \quad \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}).

The point is ((\cos \theta, \sin \theta) = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)).

Final answer: both ratios equal ( \frac{1}{\sqrt{2}} ); the point is ( \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) ).

Example 6

Simplify ( \frac{\sin \frac{\pi}{3}}{\cos \frac{\pi}{6}} ).

( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}, \quad \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}).

( \frac{\sin \frac{\pi}{3}}{\cos \frac{\pi}{6}} = 1 ).

Final answer: 1.

Where Radian Ratios Earn Their Keep

Radians tie the angle directly to arc length (s=rθ) and make calculus formulas clean.

That choice shows up wherever something oscillates or rotates:

Key Takeaways