Evaluate primary and reciprocal ratios in radians

Ontario Grade 12 Mathematics

Trigonometric Functions

Primary ratios, reciprocal ratios, and exact values from the unit circle

An angle can be measured in degrees or radians. In this lesson, every angle is in radians. You will use the unit circle to find sine, cosine, and tangent, then use those values to find their reciprocals. A calculator can also evaluate ratios for angles that do not have familiar exact values. The key is to use the correct angle mode and to check whether a denominator is zero.

What you will learn

Recognize the six primary and reciprocal trigonometric ratios.
Evaluate ratios for common radian angles using the unit circle.
Use reciprocal relationships to find cosecant, secant, and cotangent.
Identify when a trigonometric ratio is undefined.

1. Bridge from angles to the unit circle

A circle has one full turn of

2\pi

radians. This is the same as

360^\circ

. A half turn is

\pi

radians, and a quarter turn is

\frac{\pi}{2}

radians. Radians measure angles by comparing the distance along a circle with the circle’s radius.

The unit circle is a circle with radius

1

and centre at the origin of a coordinate plane. For an angle measured from the positive horizontal axis, the point on the unit circle has coordinates

(x,y)

. The horizontal coordinate is

x

and the vertical coordinate is

y

.

A primary trigonometric ratio is one of sine, cosine, or tangent. At a point

(x,y)

on the unit circle,

\cos\theta=x

and

\sin\theta=y

. When

x\ne0

,

\tan\theta=\frac{y}{x}

. These relationships let you read two ratios directly from the point and calculate the third.

\cos\theta=x,\quad \sin\theta=y,\quad \tan\theta=\frac{y}{x}

A full turn is $2\pi$ radians.
On the unit circle, cosine is the horizontal coordinate and sine is the vertical coordinate.
Tangent is the vertical coordinate divided by the horizontal coordinate, provided the horizontal coordinate is not zero.

2. Read exact values from common angles

An exact value is a value written without rounding, often using fractions or square roots. Some angles have familiar points on the unit circle. For example, the point for

\frac{\pi}{2}

is

(0,1)

, so its cosine is

0

and its sine is

1

.

The following angles are especially useful. The table lists sine and cosine; tangent can be found by dividing sine by cosine when cosine is not zero. The signs depend on the coordinates: points above the horizontal axis have positive sine, and points to its right have positive cosine.

A reciprocal ratio is found by taking

1

divided by a primary ratio. The reciprocal of sine is cosecant, written

\csc\theta

. The reciprocal of cosine is secant, written

\sec\theta

. The reciprocal of tangent is cotangent, written

\cot\theta

. A reciprocal is undefined when the ratio being inverted is zero.

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

Use the angle’s unit-circle point to determine sine and cosine.
Find tangent by dividing sine by cosine.
Find a reciprocal ratio by dividing $1$ by its corresponding primary ratio.

3. Evaluate ratios with a calculator when needed

Not every angle has a convenient exact value. A scientific calculator can approximate a ratio. Set the calculator to radian mode before entering a radian angle. If it is in degree mode, the result will correspond to a different angle.

For a calculator evaluation, enter the ratio directly, such as

\sin(1.2)

, with the angle in radians. To evaluate a reciprocal ratio, you may calculate the reciprocal of the primary ratio. Keep enough digits during intermediate calculations, then round the final result to the requested place value.

Check the denominator before calculating a reciprocal or tangent. If cosine is zero, tangent and secant are undefined. If sine is zero, cosecant and cotangent are undefined. Do not report a very large calculator output as an exact value when the ratio is actually undefined; confirm the angle or the relevant unit-circle coordinate.

\tan\theta=\frac{\sin\theta}{\cos\theta}

Choose radian mode for angles given in radians.
Round only after the calculation is complete.
A ratio with a zero denominator is undefined.

4. A reliable evaluation routine

First, identify the angle and whether the question asks for an exact value or a decimal approximation. For a familiar angle, locate its point on the unit circle. For another angle, use a calculator in radian mode.

Next, evaluate the requested primary ratio. If the question asks for a reciprocal ratio, take the reciprocal of the primary value. Keep the sign: the reciprocal of a negative number is negative.

Finally, check that the denominator is not zero. For exact answers, simplify fractions and radicals where possible. For decimal answers, follow the requested rounding instruction. These checks help prevent sign errors and undefined values.

Use exact unit-circle values when they are available.
Preserve the sign when taking a reciprocal.
Check for a zero denominator before stating a value.

Common unit-circle values

Angle	Sine	Cosine
$0$	$0$	$1$
$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$
$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$
$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$
$\frac{\pi}{2}$	$1$	$0$

Worked example

Find all six ratios at a familiar angle

Evaluate the primary and reciprocal ratios for

\theta=\frac{7\pi}{6}

.

Locate the angle
The angle $\frac{7\pi}{6}$ is $\pi+\frac{\pi}{6}$ . Its unit-circle point is in the third quadrant, where both coordinates are negative. The reference angle is $\frac{\pi}{6}$ , whose unit-circle coordinates have magnitudes $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$ .
Find sine and cosine
In the third quadrant, the horizontal and vertical coordinates are both negative. Cosine is the horizontal coordinate, and sine is the vertical coordinate.
$\cos\theta=-\frac{\sqrt{3}}{2},\quad \sin\theta=-\frac{1}{2}$
Find tangent
Tangent is sine divided by cosine. The negative signs cancel, so tangent is positive.
$\tan\theta=\frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=\frac{\sqrt{3}}{3}$
Find the reciprocal ratios
Take the reciprocal of each primary value. Rationalize the denominator in the secant result so the answer has no radical in its denominator.
$\csc\theta=-2,\quad \sec\theta=-\frac{2\sqrt{3}}{3},\quad \cot\theta=\sqrt{3}$

Answer:

\sin\theta=-\frac{1}{2},\ \cos\theta=-\frac{\sqrt{3}}{2},\ \tan\theta=\frac{\sqrt{3}}{3},\ \csc\theta=-2,\ \sec\theta=-\frac{2\sqrt{3}}{3},\ \cot\theta=\sqrt{3}

Check: The sine and cosine are both negative, so their quotient is positive. Each reciprocal multiplied by its primary ratio gives

1

.

Common mistakes and how to avoid them

Using degree mode for an angle given in radians.

Correction: Set the calculator to radian mode, or use an exact unit-circle value when possible.

Treating a reciprocal ratio as a ratio with a negative sign removed.

Correction: A reciprocal keeps the original sign. For example, the reciprocal of a negative value is negative.

Dividing by zero when evaluating tangent or a reciprocal ratio.

Correction: Check the denominator first. If it is zero, the ratio is undefined.

Confusing sine and cosine coordinates.

Correction: Cosine is the horizontal coordinate; sine is the vertical coordinate.

Lesson summary

Radians measure angles, and a full turn is $2\pi$ radians.
On the unit circle, $\cos\theta=x$ and $\sin\theta=y$ for the point $(x,y)$ .
Tangent is sine divided by cosine when cosine is not zero.
Cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent.
Use radian mode for calculator evaluations and check for zero denominators.

Check your understanding

Question 1

What is

\sin\left(\frac{3\pi}{2}\right)

?

$0$
$1$
$-1$
Undefined

Show answer and explanation

-1

At

\frac{3\pi}{2}

, the unit-circle point is

(0,-1)

. Sine is the vertical coordinate.

Question 2

If

\cos\theta=-\frac{1}{2}

, what is

\sec\theta

?

$-2$
$2$
$-\frac{1}{2}$
Undefined

Show answer and explanation

-2

Secant is the reciprocal of cosine, so

\sec\theta=\frac{1}{-\frac{1}{2}}=-2

.

Question 3

At

\theta=\pi

, which ratio is undefined?

$\sin\theta$
$\cos\theta$
$\tan\theta$
$\cot\theta$

Show answer and explanation

\cot\theta

At

\pi

, sine is

0

and cosine is

-1

. Cotangent is cosine divided by sine, so its denominator is zero. Tangent is

0

and is defined.

Key terms

Radian: A unit for measuring an angle. One full turn is $2\pi$ radians.
Unit circle: A circle with radius $1$ and centre at the origin of a coordinate plane.
Primary ratios: The trigonometric ratios sine, cosine, and tangent.
Reciprocal ratio: A ratio formed by taking $1$ divided by a primary ratio.
Undefined: A value that cannot be calculated because its expression would require division by zero.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation B1.3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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