In this page, we will will learn about the Pythagorean Identities. It is actually quite rare that exam questions are solely about these identities, but it is essential that you can use and manipulate them confidently because they are used in so many different parts of the course (so they do come up a lot!). You will learn what they are and how to use them.
The following diagram shows triangle ABC with AB = 4 and AC = 5
DIAGRAM NOT TO SCALE
a) Given that \(\large \sin \hat A=\frac{3}{4}\), find the value of \(\large \cos \hat A\)
b) Hence, show that the length of \(\large BC=\sqrt{41-10\sqrt{7}}\)
Hint
a) You can use the Pythagorean identity \(\large \cos^2\theta+\sin^2\theta\equiv1\), to find \(\large \cos \hat A\)
b) Use the Cosine Rule, c² = a² + b² - 2ab cosC
Full Solution
Question 4
Prove that \(\large \frac{\sin ^3\theta}{\tan \theta}+\cos^3\theta\equiv\cos\theta\)
Hint
This is a very difficult proof for SL students!
There is always more than one way to carry out this proof.
The easiest, is to start with the left hand side and consider that \(\large \tan\theta\equiv\frac{\sin\theta}{\cos\theta}\)
Full Solution
Question 5
a) Show that \(\large \text{cosec}^2x-\cot ^2x\equiv1\)
b) Hence, prove that \(\large \text{cosec}^4x-\cot^4x\equiv \text{cosec}^2x+\cot ^2x\)
c) Given that \(\large \text{arctan}(2)\approx63.4°\), solve
\(\large \text{cosec}^4x-\cot^4x=2-\cot x\) , for \(\large 0 \le x \le360°\)
Hint
a) Use the identity \(\large 1+\cot^{ 2 }\theta \equiv \text{cosec}^{2}\theta\)
b) Notice that the left-hand side of the identity is the difference of two squares
\(\large a^2-b^2\equiv (a-b)(a+b)\\ \)
\(\large a^4-b^4\equiv (a²-b²)(a²+b²)\\ \)
c) Use your answer in part b) to write a quadratic equation for cot x. Again, you will need the identity \(\large 1+\cot^{ 2 }\theta \equiv \text{cosec}^{2}\theta\)
Full Solution
Question 6
a) Prove that \(\large \frac{1-\tan^2x}{1+\tan^2x}\equiv\cos2x\)
b) Hence, show that
\(\large \tan\frac{\pi}{8}=\sqrt{3-2\sqrt{2}}\)
Hint
a) Start with the left-hand side of the identity.
Use \(\tan x\equiv\frac{\sin x}{\cos x}\)
b) The term hence is important here. You need to use the result proved in part a)
Try substituting \(\large x=\frac{\pi}{8}\)
Full Solution
MY PROGRESS
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