More about Trigonometric Functions: Your M2 Study Guide
Hello! Welcome to our journey into the wonderful world of advanced trigonometry. You might have thought trigonometry was just about triangles, but it's so much more! It's the language of waves, cycles, and vibrations, used in everything from music engineering to building bridges and creating video games.
In this chapter, we're going to level up our skills. We'll learn a new way to measure angles, meet three new trigonometric functions, and master a set of powerful new formulas that will be essential for calculus. Don't worry if it seems like a lot at first – we'll break it all down into simple, manageable steps. Let's get started!
1. A New Way to Measure Angles: Radians
You're used to measuring angles in degrees, where a full circle is 360°. Now, let's meet the radian, the angle measurement that mathematicians and scientists love, especially in calculus.
What on Earth is a Radian?
Imagine a circle with radius r. A radian is the angle created when you take the length of the radius and lay it along the edge (the arc) of the circle.
Analogy: Think of a pizza. If the length of the crust on your slice is exactly the same as the length from the pointy tip to the crust (the radius), the angle of your slice is 1 radian!
The formal definition is: $$ \theta (\text{in radians}) = \frac{\text{arc length}(s)}{\text{radius}(r)} $$
Since the circumference of a full circle is $$2\pi r$$, the angle for a full circle in radians is $$ \frac{2\pi r}{r} = 2\pi $$. This gives us our golden rule for conversion:
Key Conversion: $$ \pi \text{ radians} = 180^\circ $$
Converting Between Degrees and Radians
This is a skill you need to master. Just remember the key conversion!
To convert Degrees to Radians: Multiply the angle by $$ \frac{\pi}{180^\circ} $$
Example: Convert 60° to radians.
$$ 60^\circ \times \frac{\pi}{180^\circ} = \frac{60\pi}{180} = \frac{\pi}{3} \text{ radians} $$
To convert Radians to Degrees: Multiply the angle by $$ \frac{180^\circ}{\pi} $$
Example: Convert $$ \frac{3\pi}{4} $$ radians to degrees.
$$ \frac{3\pi}{4} \times \frac{180^\circ}{\pi} = \frac{3 \times 180^\circ}{4} = 3 \times 45^\circ = 135^\circ $$
Quick Review: Common Angles
- $$ 30^\circ = \frac{\pi}{6} $$
- $$ 45^\circ = \frac{\pi}{4} $$
- $$ 60^\circ = \frac{\pi}{3} $$
- $$ 90^\circ = \frac{\pi}{2} $$
- $$ 180^\circ = \pi $$
- $$ 360^\circ = 2\pi $$
Did you know?
In calculus, all the differentiation and integration formulas for trig functions (like $$(\sin x)' = \cos x$$) only work if the angle x is in radians! This is why radians are so important in advanced mathematics.
Key Takeaway for Radians
Radians are another way to measure angles, based on the radius of a circle. The most important thing to remember is the conversion factor: $$ \pi \text{ radians} = 180^\circ $$. Get comfortable converting between the two units!
2. The Reciprocal Functions: Cosecant, Secant, and Cotangent
You already know the "big three": sine, cosine, and tangent. Now, let's meet their reciprocals. They might sound fancy, but they are just the "1 over" versions of the functions you already know.
The Definitions
Cosecant (csc): The reciprocal of sine. $$ \csc\theta = \frac{1}{\sin\theta} $$
Secant (sec): The reciprocal of cosine. $$ \sec\theta = \frac{1}{\cos\theta} $$
Cotangent (cot): The reciprocal of tangent. $$ \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta} $$
Memory Aid!
It's easy to mix up which function pairs with which. Here's a trick:
Look at the third letter of the new function name.
- cosecant ($$\csc$$) pairs with sine ($$\sin$$).
- secant ($$\sec$$) pairs with cosine ($$\cos$$).
- cotangent ($$\cot$$) pairs with tangent ($$\tan$$).
The New Pythagorean Identities
You'll remember the fundamental identity: $$ \sin^2\theta + \cos^2\theta = 1 $$. We can use it to discover two new identities that are required for your syllabus.
1. Deriving the Secant-Tangent Identity:
Start with $$ \sin^2\theta + \cos^2\theta = 1 $$. Now, divide every term by $$ \cos^2\theta $$.
$$ \frac{\sin^2\theta}{\cos^2\theta} + \frac{\cos^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta} $$
This simplifies to:
$$ \tan^2\theta + 1 = \sec^2\theta $$
2. Deriving the Cosecant-Cotangent Identity:
Start with $$ \sin^2\theta + \cos^2\theta = 1 $$. This time, divide every term by $$ \sin^2\theta $$.
$$ \frac{\sin^2\theta}{\sin^2\theta} + \frac{\cos^2\theta}{\sin^2\theta} = \frac{1}{\sin^2\theta} $$
This simplifies to:
$$ 1 + \cot^2\theta = \csc^2\theta $$
Quick Review: The 3 Pythagorean Identities
- $$ \sin^2\theta + \cos^2\theta = 1 $$
- $$ 1 + \tan^2\theta = \sec^2\theta $$
- $$ 1 + \cot^2\theta = \csc^2\theta $$
These are incredibly useful for simplifying complicated trigonometric expressions!
Key Takeaway for Reciprocal Functions
Cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent. Memorize their definitions and the two new Pythagorean identities. They are key tools for simplifying expressions.
3. Adding and Subtracting Angles: Compound Angle Formulae
What is the value of $$ \sin(A+B) $$? It's a common mistake to think it's just $$ \sin A + \sin B $$. Let's prove that's wrong:
Let A=30° and B=60°.
$$ \sin(30^\circ+60^\circ) = \sin(90^\circ) = 1 $$.
But $$ \sin(30^\circ) + \sin(60^\circ) = 0.5 + \frac{\sqrt{3}}{2} \approx 1.366 $$.
They are not equal!
To find the right answer, we need the Compound Angle Formulae. These are must-know formulas for your exam.
Formulae List: Compound Angles
- $$ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B $$
(The sign in the middle stays the same.)
- $$ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B $$
(The sign in the middle is opposite!)
- $$ \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} $$ (The top sign is the same, the bottom sign is opposite.)
How to Use Them: Step-by-Step
Example: Find the exact value of $$ \sin(75^\circ) $$ without a calculator.
Step 1: Break the angle into two "special" angles that you know the trig values for (like 30°, 45°, 60°).
We can write $$ 75^\circ = 45^\circ + 30^\circ $$.
Step 2: Choose the correct compound angle formula.
We need $$ \sin(A+B) $$, so we use $$ \sin(A+B) = \sin A \cos B + \cos A \sin B $$.
Step 3: Substitute your angles and their known values.
$$ \sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ) $$ $$ = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) $$
Step 4: Simplify the expression.
$$ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6}+\sqrt{2}}{4} $$
Key Takeaway for Compound Angles
These formulae let you find the trig function of a sum or difference of angles. Memorize the three main formulas and pay close attention to the signs ($$\pm$$ and $$\mp$$).
4. Doubling Up: The Double Angle Formulae
The Double Angle Formulae are just a special case of the compound angle formulae where A = B. They are so useful they get their own name!
Deriving the Formulae
Let's find $$ \sin(2A) $$. We use $$ \sin(A+B) $$ and set B = A.
$$ \sin(2A) = \sin(A+A) = \sin A \cos A + \cos A \sin A = 2\sin A \cos A $$
Easy, right? Let's do the same for cosine and tangent.
Formulae List: Double Angles
- $$ \sin(2A) = 2\sin A \cos A $$
- $$ \cos(2A) = \cos^2 A - \sin^2 A $$
This one has two other very useful forms! By substituting $$ \cos^2 A = 1 - \sin^2 A $$ or $$ \sin^2 A = 1 - \cos^2 A $$, we get:
- $$ \cos(2A) = 1 - 2\sin^2 A $$
- $$ \cos(2A) = 2\cos^2 A - 1 $$
- $$ \tan(2A) = \frac{2\tan A}{1 - \tan^2 A} $$
From Double Angle to Power Reduction
By rearranging the formulae for $$ \cos(2A) $$, we can get expressions for $$ \sin^2 A $$ and $$ \cos^2 A $$. These are very important for calculus (especially integration!). The syllabus requires you to know them.
Formulae List: Power Reduction
- From $$ \cos(2A) = 1 - 2\sin^2 A \implies \sin^2 A = \frac{1}{2}(1 - \cos(2A)) $$
- From $$ \cos(2A) = 2\cos^2 A - 1 \implies \cos^2 A = \frac{1}{2}(1 + \cos(2A)) $$
Key Takeaway for Double Angles
These are shortcuts for finding the trig values of doubled angles (like $$ \theta \to 2\theta $$). Master the three forms of $$ \cos(2A) $$ and the derived power reduction formulas, as they are essential problem-solving tools.
5. Transforming Expressions: Product-to-Sum and Sum-to-Product
These last two sets of formulae might look intimidating, but they are just tools for changing the *form* of a trig expression. Sometimes it's easier to work with a sum, and other times a product is better. These formulae let you switch between the two.
Product-to-Sum Formulae
Use these to turn a product of sine/cosine functions into a sum or difference.
- $$ 2\sin A \cos B = \sin(A+B) + \sin(A-B) $$
- $$ 2\cos A \cos B = \cos(A+B) + \cos(A-B) $$
- $$ 2\sin A \sin B = \cos(A-B) - \cos(A+B) $$ (Be careful with this one! The order is A-B then A+B).
Example: Express $$ 2\sin(4x)\cos(2x) $$ as a sum.
Using the first formula with A=4x and B=2x:
$$ 2\sin(4x)\cos(2x) = \sin(4x+2x) + \sin(4x-2x) = \sin(6x) + \sin(2x) $$
Sum-to-Product Formulae
Use these to turn a sum or difference into a product.
- $$ \sin A + \sin B = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) $$
- $$ \sin A - \sin B = 2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) $$
- $$ \cos A + \cos B = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) $$
- $$ \cos A - \cos B = -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) $$ (Watch out for the negative sign on this last one!)
Example: Express $$ \cos(7\theta) + \cos(\theta) $$ as a product.
Using the third formula with A=7θ and B=θ:
$$ \cos(7\theta) + \cos(\theta) = 2\cos\left(\frac{7\theta+\theta}{2}\right)\cos\left(\frac{7\theta-\theta}{2}\right) = 2\cos(4\theta)\cos(3\theta) $$
Key Takeaway for Transformations
These formulas are all about changing the form of an expression. You don't need to derive them in an exam, but you do need to recognize when to use them and how to apply them correctly. Keep this list handy when you practice!