Question

Find the exact value of sec(–135°).

Answer 1

Answer: $-\sqrt{2}$

Step-by-step explanation:

The trigonometric function secant is the reciprocal of cosine:

$sec(\alpha)=\frac{1}{cos(\alpha)}$

For $-135\°$:

$sec(-135\°)=\frac{1}{cos(-135\°)}$ (1)

On the other hand, it is known $cos(-\alpha)=cos(\alpha)$, hence:

$cos(-135\°)=cos(135\°)$ (2)

In addition, it is known $cos(135\°)=-cos(45\°)=-\frac{\sqrt{2}}{2}$ (3)

Substituting this on (1):

$sec(-135\°)=\frac{1}{-\frac{\sqrt{2}}{2}}$ (4)

$sec(-135\°)=-\frac{2}{\sqrt{2}}$ (5)

$sec(-135\°)=-\frac{2}{\sqrt{2}}(\frac{\sqrt{2}}{\sqrt{2}})$ (6)

Finally:

$sec(-135\°)=-\sqrt{2}$