Question

Without using tables, find the value of tan 450

A 3
B 5
C 2
D 1

Answer 1

Answer on Question #51810 – Math – Trigonometry

Question

Without using tables, find the value of tan 450

A 3

B 5

C 2

D 1

Solution

Take the unit circle, the triangle is right-angled, therefore $x = \cos(\alpha)$,

$y = \sin(\alpha)$ and $x^2 + y^2 = 1$. Besides, the triangle is isosceles, therefore

$x = y$.

If the triangle is right-angled and isosceles, then its interior angles are $45{}^\circ$, $45{}^\circ$, $90{}^\circ$.

Solving the system of equations

$$\left\{ \begin{array}{c} x^2 + y^2 = 1, \\ x = y, \end{array} \right.$$

substitute $y = x$ and the first equation gives

$$x^2 + x^2 = 1$$ $$2x^2 = 1$$ $$x = \sqrt{1/2}$$ $$x = \frac{\sqrt{2}}{2}.$$

Thus, $x = y = \frac{\sqrt{2}}{2}$. In other words, $\cos(45{}^\circ) = \sin(45{}^\circ) = \frac{\sqrt{2}}{2}$.

$$\tan 45{}^{\circ} = \frac{\sin 45{}^{\circ}}{\cos 45{}^{\circ}} = \frac{\sqrt{2}}{\sqrt{2}} = 1;$$

Answer: D 1.

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