Trigonometric Functions (Cambridge International IGCSE Further Maths)

Trigonometric functions

What are the trigonometric functions?

• The three standard trigonometric functions are: sine, cosine and tangent

• The functions are many-to-one mappings

  • They do not have inverses unless their domain is restricted

How can I use the trigonometric functions with right-angled triangles?

• The three trigonometric functions sine, cosine and tangent give the ratios of side lengths in right-angled triangles

• For an acute angle in a right-angled triangle:

  • Sine of the angle is the length of the side opposite the angle divided by the hypotenuse
  • Cosine of the angle is the length of the side adjacent to the angle divided by the hypotenuse
  • Tangent of the angle is the length of the opposite side divided by the length of the adjacent side

• You can use the acronym SOHCAHTOA to help you remember these

How can I use the trigonometric functions with non-right-angled triangles?

• You can find missing angles and the lengths of missing sides using the sine rule and the cosine rule
• You can also use the area formula

Reciprocal Trig Functions

What are the reciprocal trig functions?

• There are three reciprocal trig functions that each correspond to either sin, cos or tan

  • Secant(sec x)

    • $\sec x = \frac{1}{\cos x}$

  • Cosecant(cosec x)

    • $\text{cosec} x = \frac{1}{\sin x}$

  • Cotangent (cot x)

    • $\cot x = \frac{1}{\tan x }$

• A good way to remember which function is which is to look at the third letter in each of the reciprocal trig functions

  • cot x is 1 over tan x etc

• Each of the reciprocal trig functions are undefined for certain values of x

  • sec x is undefined for values of x* for which cos x = 0

  • cosec x is undefined for values of x for which sin x = 0

  • cot x is undefined for values of x* for which tan x = 0

    • When tan x is undefined, cot x = 0

• Be careful not to confuse the reciprocal trig functions with the inverse trig functions

  • $\sin ^{- 1} x \text{eq} \frac{1}{\sin x}$

Worked Example Without the use of a calculator, find the values of

a) $\sec \frac{\pi }{6}$

The third letter of sec is c so it is the reciprocal of cos.

sec ( pi / 6 ) = (1)/(cos ( pi / 6 ))

Write down the value of cos ( pi / 6 ).

sec ( pi / 6 ) = (1)/(( (sqrt( 3)/(2) ))

Simplify.

$\mathrm{sec}(\frac{π}{6})=\frac{2}{\sqrt{3}}\mathrm{or}\frac{2\sqrt{3}}{3}$

b) $\mathrm{cot}45^{\circ}$

The third letter of cot is t so it is the reciprocal of tan.

$\cot ( 45^{\circ} ) = \frac{1}{\tan ( 45^{\circ} )}$

Write down the value of tan(45°).

$\cot ( 45^{\circ} ) = \frac{1}{1}$

$\mathrm{cot}(45^{\circ})=1$