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Exact trigonometric values

Exact trigonometric values

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Tutor: Bilal

Summary

Exact trigonometric values

In a nutshell

While most trigonometric values are very hard (or downright impossible) to find without using a calculator, there are very specific values that can be memorised.



Trigonometry values to memorise


xx​​

sin(x)\sin(x)​​

cos(x)\cos(x)​​​

tan(x)\tan(x)​​

00​​

00​​

11​​

00​​

3030​​

12\frac{1}{2}​​

32\frac{\sqrt{3}}{2}​​

13=33\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}​​

4545​​

12=22\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}​​

12=22\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}​​

11​​

6060​​

32\frac{\sqrt{3}}{2}​​​

12\frac{1}{2}​​​

3\sqrt{3}​​

9090​​

​​11​​

00​​

n/an/a



An easier way to learn the values

This may seem like a lot to memorise, but there are easier ways to remember these. The table below is easy to construct when you realise the numbers are consecutive (0,1,2,3,40,1,2,3,4​). To use this table:


Maths; Trigonometry; KS4 Year 10; Exact trigonometric values


Procedure

​​1.

Locate the number associated with the trigonometric function and angle you want.

2.

Take the number, square root it, and then divide by 2 (hence the design of the table).

Example 1

What is the value of cos(60)\cos(60)?


Refer to the table to see that cos(60)\cos(60) corresponds to the number 1.


Maths; Trigonometry; KS4 Year 10; Exact trigonometric values


Hence, square root this value, and then halve it:


cos(60)=12=12\underline{\cos(60) = \frac{\sqrt{1}}{2} = \frac{1}{2}}


What about tan()\tan()​?

The table only gives values for sin()\sin() and cos()\cos(). The easiest way to learn values for tan()\tan() is to just memorise the four values. Alternatively, you can find the value for sin()\sin() and cos()\cos() of the particular angle and divide them.


Example 2

What is the value of tan(30)\tan(30)?


First find sin(30)\sin(30) by referring to the table. The value in the table is 11, giving

sin(30)=12\sin(30)= \dfrac 1 2 ​​


Then find cos(30)\cos(30), using the same method, giving

cos(30)=32\cos(30)=\dfrac {\sqrt3}{2}​​


Find the value of tan(30)\tan(30) by dividing sin(30)\sin(30) by cos(30)\cos(30)

tan(30)=sin(30)cos(30)=1232=13=33\tan(30) = \dfrac{\sin(30)}{\cos(30)} = \dfrac{\frac 1 2}{\frac {\sqrt3}{2}} = \dfrac {1}{\sqrt3}=\underline{\dfrac{\sqrt3}{3}}​​

​​​

Common trigonometric value problems

You should know how to apply the common trigonometric values to standard trigonometry with a right-angled triangle.


Example 3

Find the exact value of xx in the diagram below.

Maths; Trigonometry; KS4 Year 10; Exact trigonometric values


To solve this, treat this like a normal trigonometric question.

Label the diagram:

Maths; Trigonometry; KS4 Year 10; Exact trigonometric values


The adjacent and hypotenuse are the important lengths here. So, use CAH:

cos(x)=AH\cos(x)=\frac{A}{H}​​


Substitute in values:

cos(45)=10x\cos(45)=\frac{10}{x}​​


Rearrange to solve for the hypotenuse:

x=10cos(45)x=\frac{10}{\cos(45)}​​


Use the common trigonometric ratios to find the value of cos(45)\cos(45):

cos(45)=12\cos(45)=\frac{1}{\sqrt{2}}​​


Substitute this value in to find the exact value of xx:

x=1012x=\frac{10}{\frac{1}{\sqrt{2}}}​​


x=102cm\underline{x=10\sqrt{2}cm}​​


Note: When a question asks for you to find the exact value, it means that you should leave your answer as a surd or fraction.



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FAQs - Frequently Asked Questions

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