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Trigonometric Functions (CIE IGCSE Additional Maths)

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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
Function Notation Domain Range
Sine sin open parentheses x close parentheses x element of straight real numbers negative 1 less or equal than sin open parentheses x close parentheses less or equal than 1
Cosine cos open parentheses x close parentheses x element of straight real numbers negative 1 less or equal than cos open parentheses x close parentheses less or equal than 1
Tangent tan open parentheses x close parentheses x element of straight real numbers x not equal to plus-or-minus 90 degree comma plus-or-minus 270 degree comma plus-or-minus 450 degree comma... space open parentheses in space degrees close parentheses x not equal to plus-or-minus straight pi over 2 comma plus-or-minus fraction numerator 3 straight pi over denominator 2 end fraction comma plus-or-minus fraction numerator 5 straight pi over denominator 2 end fraction comma... space open parentheses in space radians close parentheses tan open parentheses x close parentheses element of straight real numbers

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

The trig functions in relation to the sides of a right-angled triangle 

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

The formula for the sine rule, cosine rule and area of a triangle

 

Non-Right-Angled Triangles Diagram 1b, A Level & AS Level Pure Maths Revision Notes

 

Non-Right-Angled Triangles Diagram 1c, A Level & AS Level Pure Maths Revision Notes

 

Non-Right-Angled Triangles Diagram 1d, A Level & AS Level Pure Maths Revision Notes

 

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 invisible function application x equals fraction numerator 1 over denominator cos space invisible function application x end fraction
    • Cosecant (cosec x)
      • cosec blank x equals fraction numerator 1 over denominator sin space invisible function application x end fraction
    • Cotangent (cot x)
      • cot space x blank equals fraction numerator 1 over denominator tan invisible function application space x blank end fraction  
  • 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 to the power of negative 1 end exponent invisible function application space x blank not equal to fraction numerator 1 over denominator sin space invisible function application x end fraction

Worked example

Without the use of a calculator, find the values of

a)
sec space invisible function application pi over 6

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

sec open parentheses straight pi over 6 close parentheses equals fraction numerator 1 over denominator cos open parentheses straight pi over 6 close parentheses end fraction

Write down the value of cos open parentheses straight pi over 6 close parentheses.

sec open parentheses straight pi over 6 close parentheses equals fraction numerator 1 over denominator open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses end fraction

Simplify.

bold sec stretchy left parenthesis pi over 6 stretchy right parenthesis bold equals fraction numerator bold 2 over denominator square root of bold 3 end fraction bold space bold or bold space fraction numerator bold 2 square root of bold 3 over denominator bold 3 end fraction

b)
cot space 45 degree 

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

cot open parentheses 45 degree close parentheses equals fraction numerator 1 over denominator tan open parentheses 45 degree close parentheses end fraction

Write down the value of tan(45°).

cot open parentheses 45 degree close parentheses equals 1 over 1

bold cot stretchy left parenthesis 45 degree stretchy right parenthesis bold equals bold 1

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.