Co-functions and Special Angles

Q: When do I use co-functions?

When angles add up to 90°, or when you see $90° \pm \theta$. Sine and cosine swap.

Q: Is the square identity used with co-functions?

Often. $\sin^2 20° + \sin^2 70°$ becomes $\sin^2 20° + \cos^2 20° = 1$.

Co-functions link sine and cosine through 90°, and special angles give exact values without a calculator. Together with reduction formulae they make up the 'simplify without a calculator' questions that appear in every Paper 2.

Co-function identities with 90°

In a right-angled triangle, the two acute angles add up to 90°. The side opposite one is adjacent to the other, so

$sin (90° - θ) = cos θ cos (90° - θ) = sin θ$

For $90° + θ$ (quadrant 2, where only sine is positive):

$sin (90° + θ) = cos θ cos (90° + θ) = - sin θ$

Example. $\frac{sin 70°}{cos 20°} = \frac{cos 20°}{cos 20°} = 1$

Look for pairs of angles that add up to 90°, such as 20° and 70° or 35° and 55°.

Special angles without a calculator

From the 30°-60°-90° triangle (sides 1, $3$ , 2) and the 45° triangle (sides 1, 1, $2$ ):

$sin 30° = \frac{1}{2}$ , $cos 30° = \frac{3}{2}$ , $tan 30° = \frac{1}{3}$
$sin 45° = cos 45° = \frac{1}{2}$ , $tan 45° = 1$
$sin 60° = \frac{3}{2}$ , $cos 60° = \frac{1}{2}$ , $tan 60° = 3$
$sin 0° = 0$ , $cos 0° = 1$ , $sin 90° = 1$ , $cos 90° = 0$ , $tan 90°$ undefined

Example. $\frac{tan 60° \cdot cos 30°}{sin 45° \cdot cos 45°} = \frac{3 \cdot \frac{3}{2}}{\frac{1}{2}} = 3$

Draw the two triangles on your page at the start of the exam. It takes seconds and prevents errors.

Key terms

Co-function: A pair of ratios linked through $90° - θ$ , such as sine and cosine.
Complementary angles: Two angles that add up to 90°.
Exact value: A value written with fractions and surds rather than decimals.

Common mistakes

Applying co-functions to $180° \pm θ$ .
Mixing up $sin 30°$ and $sin 60°$ .
Getting the sign wrong for $cos (90° + θ)$ .
Using a calculator when the question says not to.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

Simplify without a calculator: $\frac{sin 70°}{cos 20°}$

Worked solution

Hidden - try the question first

Solution

$sin 70° = sin (90° - 20°)$
$= cos 20°$
$\frac{cos 20°}{cos 20°} = 1$

Question 2

5 marks total

routine

Without using a calculator, determine the value of $\frac{tan 60° \cdot cos 30°}{sin 45° \cdot cos 45°}$ .

Worked solution

Hidden - try the question first

Solution

$sin 70° = sin (90° - 20°)$ and $cos 30° = \frac{3}{2}$
$sin 45° \cdot cos 45° = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{2}$
Numerator $= \frac{3}{2}$
$\frac{\frac{3}{2}}{\frac{1}{2}}$
$= 3$

Question 3

6 marks total

complex

Simplify fully: $\frac{sin ( 90° + x ) \cdot cos ( 90° - x )}{cos ( 180° - x ) \cdot tan ( - x )}$

Worked solution

Hidden - try the question first

Solution

$sin 70° = sin (90° - 20°)$
$= cos 20°$
$cos (180° - x) = - cos x$ and $tan (- x) = - tan x$
$\frac{cos x sin x}{( - cos x ) ( - tan x )} = \frac{sin x}{tan x}$
$= sin x \cdot \frac{cos x}{sin x}$
$= cos x$

Question 4

8 marks total

problem-solving

Without using a calculator, determine the value of:

(a) $sin^{2} 20° + sin^{2} 70° + tan 45° \cdot cos 60°$ (4)

(b) $\frac{sin 35°}{cos 55°} + cos^{2} 15° + cos^{2} 75°$ (4)

Worked solution

Hidden - try the question first

Solution

(a) $sin 70° = cos 20°$
$sin^{2} 20° + cos^{2} 20° = 1$
$tan 45° \cdot cos 60° = 1 \times \frac{1}{2}$
$1 + \frac{1}{2} = \frac{3}{2}$
(b) $cos 55° = sin 35°$ , so the fraction is 1
$cos 75° = sin 15°$
$cos^{2} 15° + sin^{2} 15° = 1$
$1 + 1 = 2$

FAQs

When do I use co-functions?

When angles add up to 90°, or when you see $90° \pm θ$ . Sine and cosine swap.

Do I have to memorise the special angle values?

Yes, or learn to draw the two special triangles quickly. They are not on the formula sheet.

Is the square identity used with co-functions?

Often. $sin^{2} 20° + sin^{2} 70°$ becomes $sin^{2} 20° + cos^{2} 20° = 1$ .

Co-function identities with 90°

Special angles without a calculator

Key terms

Common mistakes

Practice questions

FAQs

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