Zion Robles

2023-03-15

How to evaluate $\mathrm{sec}110\xb0$?

progutannx7f

Beginner2023-03-16Added 7 answers

We given explanation: $\mathrm{sec}110=\frac{1}{\mathrm{cos}110}$. Find cos 110 without using calculator.

Call cos (110) = cos x

Use this equation: $\mathrm{cos}3x=4{\mathrm{cos}}^{3}x-3\mathrm{cos}x.$ We get:

cos 330 = cos (-30 + 360) = cos 30 = sqrt3/2

$\mathrm{cos}330=\frac{\sqrt{3}}{2}=4{\mathrm{cos}}^{3}x-3\mathrm{cos}x$

$4{\mathrm{cos}}^{3}x-3\mathrm{cos}x-\frac{\sqrt{3}}{2}=0$

Use a graphing calculator to solve this cubic problem.

By estimation, we get;

cos x = - 0.34 --> x = 110 (accepted) --> 3x = 330

cos x = - 0.65 --> x = 130 (rejected) --> 3x = 390

cos x = 0.98 ----> x = 10 (rejected) -->3x = 30

Hence:

$\mathrm{sec}\left(110\right)=\frac{1}{-0.34}=-2.94$

graph{4x^3 - 3x - sqrt3/2 [-2.5, 2.5, -1.25, 1.25]}

Call cos (110) = cos x

Use this equation: $\mathrm{cos}3x=4{\mathrm{cos}}^{3}x-3\mathrm{cos}x.$ We get:

cos 330 = cos (-30 + 360) = cos 30 = sqrt3/2

$\mathrm{cos}330=\frac{\sqrt{3}}{2}=4{\mathrm{cos}}^{3}x-3\mathrm{cos}x$

$4{\mathrm{cos}}^{3}x-3\mathrm{cos}x-\frac{\sqrt{3}}{2}=0$

Use a graphing calculator to solve this cubic problem.

By estimation, we get;

cos x = - 0.34 --> x = 110 (accepted) --> 3x = 330

cos x = - 0.65 --> x = 130 (rejected) --> 3x = 390

cos x = 0.98 ----> x = 10 (rejected) -->3x = 30

Hence:

$\mathrm{sec}\left(110\right)=\frac{1}{-0.34}=-2.94$

graph{4x^3 - 3x - sqrt3/2 [-2.5, 2.5, -1.25, 1.25]}

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