Pythagorean Identity
Let's start with this triangle again:
Let's label the sides $o$, $a$, and $h$ for brevity.
Now, with the Pythagorean theorem, we can write:
$$o^2 + a^2 = h^2$$
Dividing both sides by $h^2$ we get:
$$\frac{o^2}{h^2} + \frac{a^2}{h^2} = \frac{h^2}{h^2} \to (\frac{o}h)^2+(\frac{a}h)^2=1$$
But $\sin\theta = \frac{o}h$ and $\cos\theta = \frac{a}h$. So, the above simplifies to:
$\cos^2 θ + \sin^2 θ = 1$
This is the Pythagorean identity in trigonometry.
The Pythagorean identity holds true for all real values of θ, not just acute angles.
Let's verify the Pythagorean identity for θ = 30°:
$\cos^2 30° + \sin^2 30° = (\frac{\sqrt{3}}{2})^2 + (\frac{1}{2})^2 = \frac{3}{4} + \frac{1}{4} = 1$
Double Angle Identities
Double angle identities express trigonometric functions of 2θ in terms of functions of θ. The two most important double angle identities are for sine and cosine:
- $\sin 2θ = 2\sin θ \cos θ$
- $\cos 2θ = \cos^2 θ - \sin^2 θ = 2\cos^2 θ - 1 = 1 - 2\sin^2 θ$
There are three equivalent forms for cos 2θ. Choosing the most appropriate form can simplify calculations significantly.
Using the Pythagorean identity, you can derive alternate formulas for $\cos2\theta$:
- $\cos2\theta = 1-2\sin^2\theta$
- $\cos2\theta = 2\cos^2\theta + 1$
These could save you a bit of time on an exam.
These identities can be derived using complex numbers. (You'll know what they are if you take HL, otherwise don't worry about them.)
Let's use the double angle identity to verify sin 60°:
We know that 60° is double 30°, so:
$\sin 60° = \sin 2(30°) = 2\sin 30° \cos 30° = 2 \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}$
Problem-Solving Without Finding Specific Angles
With the Pythagorean identity and double angle identites, you can find trigonometric results without actually finding the angle itself. This is useful in exams, particularly paper 1 where you don't have a calculator.
Given cos x = 3/4 and x is acute, find sin 2x without finding x:
We can use the double angle identity for sine: $\sin 2x = 2\sin x \cos x$
We know cos x = 3/4. To find sin x, we use the Pythagorean identity: $\sin^2 x + (\frac{3}{4})^2 = 1$ $\sin^2 x = 1 - \frac{9}{16} = \frac{7}{16}$ $\sin x = \frac{\sqrt{7}}{4}$ (positive as x is acute)
Now we can calculate: $\sin 2x = 2 \cdot \frac{\sqrt{7}}{4} \cdot \frac{3}{4} = \frac{3\sqrt{7}}{8}$
Students often try to find the angle x first and then use it to calculate sin 2x. This approach is more time-consuming and prone to rounding errors. Using identities directly is usually more efficient and accurate.
