Trigonometric Identities
General Ideas of Modelling in Maths - including "word problems", an introduction only! More to come...

Trigonometric Identities

The trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant, and their inverses arcsine etc.) relate to each other in many different ways. These relations form trigonometric identities: equations which involve trigonometric functions and hold true for any value of the variable in the equation.

It is worth being familiar with some of the trigonometric identities, but it isn't necessary to memorise them all. Check out our ASK Poster here.

The ones we suggest are worth "memorising" include:

tan A = sin A /cos A Yes, this is a Trig Identity. And it falls out of the definitions of sin, cos and tan.
cos² A + sin² A = 1 This is known as the Pythagorean Identity as it falls straight out of Pythagoras' Theorem.
sin(A+B) = sin A cos B + cos A sin B and cos(A+B) = cos A cos B - sin A sin B

All other identities are either derived relatively easily from these above, e.g. the Double Angle Formulae includes sin (2A) = 2 sin A cos A, or they are pretty obscure and so best looked up.

You should also know that any sum of trigonometric functions can be eventually expressed as a single trigonometric function. This becomes obvious if you have found out about Fourier series.

Resources

ASK Trigonometry Overview

Basic Trigonometry - Further ASK Resources on trigonometry.

Definition of Identity | mathsisfun.com

First Aid - Trigonometrical Identities - A reference list of the most usual trig ids.

Trig Identities ASK Poster

Summary of Trigonometric Identities | clarku.edu

Working with Trigonometric Relationships in Degrees | BBC Bitesize

First Aid - The Sine and Cosine Rule Worksheet (Worksheet with Answers)

Mathcentre - Expressing 2 Trig Functions as 1 Worksheet (Worksheet with Answers)

Solving Trigonometric Equations With Identities | Precalculus (lumenlearning.com) - Hints and tips on verifying trig ids.

Try this...

Have a go at the questions shown below. The answers for questions 1 and 2 are linked here.

Use Pythagoras' Theorem to show that $LaTeX: \sin^2\theta+\cos^2\theta=1$ for all angles $LaTeX: \theta$ .
Apply simple trigonometric identities to the following problems:
1. If $LaTeX: \tan A=\frac{1}{3}$ what is the value of $LaTeX: \cot A$ ?
2. If $LaTeX: \cos\left(\frac{\pi}{5}\right)=0.8090$ , what is the value of $LaTeX: \sin\left(\frac{3\pi}{10}\right)$ (NO calculators! Hint: apply one of the "co-function" identities in this poster: Trig Identities ASK Poster)
3. If $LaTeX: \sin B=0.5$ what are the possible values of $LaTeX: \cos B$ ?
4. If $LaTeX: \sin C=0.2$ what is the value of $LaTeX: \sin2C$ ?
5. If $LaTeX: \cos B=0.1$ what is the value of $LaTeX: \cos\frac{B}{2}$ ?
6. If $LaTeX: \sin A=\frac{3}{5}$ and A is an acute angle, use the double angle formulas to find exact values of $LaTeX: \sin2A$ , $LaTeX: \cos2A$ , $LaTeX: \tan2A$ . In which quadrant is angle ?
A worksheet! 10 pages - do pick and choose from them: More Trigonometric Identities Worksheet | uky.edu
Try plotting some trig ids on GeoGebra. E.g. for the identity $LaTeX: \tan\left(\alpha+\beta\right)=\frac{\tan\:\alpha+\:\tan\beta}{1-\:\tan\alpha\:\tan\:\beta}$ , first plot (say) $LaTeX: y=\tan\left(x+2.5\right)$ and then $LaTeX: y=\frac{\tan\:x+\:\tan2.5}{1-\:\tan x\:\tan\:2.5}$ , and check out a few more values for $LaTeX: \beta$ . You can try the 3D graphing feature in Geogebra to see the full effect.

More...

First Aid - Hyperbolic Functions Worksheet (Worksheet with Answers)

First Aid - Hyperbolic Identities List

Ideas of Modelling with Maths

In real life, we are often faced with a problem that we might wish to solve or analyse to understand better. Mathematics gives us the tools to seek patterns and explore how quantities and other types of parameter vary with each other. Mathematical modelling is the process of taking a real problem and creating a mathematical expression that in some way describes the problem, and which lets us apply maths to that problem.

At school, you may have come across simple "word problems" and been asked to convert them to maths. For example: Ali has 10 apples and Ali and Bertie together have 13 apples: how might you write an expression that shows this and which may then let you calculate the number of apples that Bertie has? This is in fact the start of modelling. Engineers spend a lot of time either working with mathematical models in the form of formulae/equations/mathematical constructs that link variables together, or to create mathematical models.

The basic and outline procedural steps in modelling with maths are:

Identify what the question is!
What are the variables? What are the givens?
Find an expression that links the variables. You may need to apply known laws OR you may use experimental data and graphs. This will be your MODEL.
Check that the model against the data you have and new cases.
Correct/simplify/adjust - goto step 1.

This image (from (Yong, Levy & Lape, 2015), helps to understand how modelling is an interpretation of the real world and includes many more steps:

Yong; D., Levy, R; Lape N., Why No Difference? A Controlled Flipped Classroom Study for an Introductory Differential Equations Course, November 2015, PRIMUS: problems, resources, and issues in mathematics undergraduate studies 25(9-10):907-921
DOI:10.1080/10511970.2015.1031307 (from https://www.researchgate.net/publication/283805990_Why_No_Difference_A_Controlled_Flipped_Classroom_Study_for_an_Introductory_Differential_Equations_Course accessed 26 April, 2022)

To be able to do mathematical modelling, you need to be fluent in algebra. In engineering, much modelling involves differential equations. Hint and tip: get comfortable with the language and concepts of calculus, i.e. the idea of rate of change!

More to come...

Resources

Mathematical model - Wikipedia

Real World Examples of Mathematical Modelling - Maths Careers - Intro by IMA

What Is Mathematical Modelling? - Maths Careers - more by the IMA on mathematicians at work.

Refer back to Mathematics - Basics - Introductory Algebra & Mathematics - Introduction to Calculus of course, and also Mathematics - Applying Calculus.