Solving Trigonometric Questions Without a Calculator

Question

How do I solve the following question without using a calculator?

Answer 1

• Find quickly the point of intersection $S(3,4)$
• Note that the triangle $P(1,0),Q(0,1),S(3,4)$ has a right angle at $Q$.
• Hence, $\tan \alpha = \frac{|PQ|}{|QS|} = \frac{\sqrt{2}}{\sqrt{2}\cdot 3} = \frac{1}{3}$

Answer 2

Hint: Find the equation of two lines using the information(marked coordinates) given in the problem, the coefficient of $x$ are the slopes. Suppose the slope of the line passing the rightmost point of $X$-axis is $m_1$ and of the other line $m_2$. Then $$\tan\alpha=\Big(\frac{m_1-m_2}{1+m_1m_2}\Big)$$

Answer 3

You can notice that the directing vectors of the 2 line are $\vec v_1(-1,-1,0)$ and $\vec v_2(-1,-2,0)$

From the cross product you get that $\sin (\alpha)=\frac{1}{\sqrt{10}}$

From the dot product you get that $\cos (\alpha)=\frac{3}{\sqrt{10}}$

Divide them to get $$\tan (\alpha)=\frac{1}{3}$$