Express the length a, b, c, and d in the figure in terms of the trigonometric ratios of θ.

Question

Problem

Express the length a, b, c, and d in the figure in terms of the trigonometric ratios of $θ$. (See the image below)

Progress

I can figure out $c$ usng the pythagorean theorem. $a^2+b^2=c^2$ which would be $2$. Is that correct? How do I solve the rest?

---

Original image

Answer 1

Compute $\sin(\theta)$, $\cos(\theta)$ and $\tan(\theta)$ using your picture. What do you see?

Answer 2

I have marked 5 points P,Q,R,S,T in your diagram.

Consider the triangle formed by points P, S and T. Do you agree that

• Length of PS = c
• Length of ST = b
• Length of PT = 1

It is a right-angle triangle, so apply Pythagoras theorem,

square of the hypotenuse is equal to the sum of the squares of the other two sides

We get $b^2+1^2=c^2$

We know that $\sin \theta=\Large \frac{\text{opposite side}}{\text{hypotenuse}}$and $\cos\theta = \Large \frac{\text{adjacent}}{\text{hypotenuse}}$(Now, refer triangle PST and write down write $\sin\theta,\cos\theta$)

Now, simplify, substitue to express b and c only in terms of trig ratio, that is b=some tri ratio($\theta$) and c=some trig ratio($\theta$)

Similarly, for triangle PRQ Do you agree that

• Length of PQ = d
• Length of RQ = a
• Length of PR = 1