Trigonometry - Sec, Csc, Ctan

Find the value of the trigonometric function in fraction form for triangle .

Triangle

What is the secant of $\angle A$ ?

$\frac{25}{24}$

Explanation

The value of the secant of an angle is the value of the hypotenuse over the adjacent.

Therefore:

$sec \angle A = \frac{hypotenuse}{adjacent} = \frac{25}{24}$

Determine the value of $cot(\frac{\pi}{4})$ .

$\infty$

$\frac{1}{2}$

$\frac{\sqrt2}{2}$

Explanation

Rewrite $cot(\frac{\pi}{4})$ in terms of sine and cosine.

$cot(\frac{\pi}{4})= \frac{cos(\frac{\pi}{4})}{sin(\frac{\pi}{4})}=\frac{\frac{\sqrt2}{2}}{\frac{\sqrt2}{2}}=1$

Which of the following is the equivalent to $\frac{1}{\csc\theta}$ ?

$\sin\theta$

$\cos\theta$

$\sec\theta$

$\tan\theta\sin\theta$

$\cot\theta$

Explanation

Since $\csc\theta=\frac{1}{\sin\theta}$ :

$\frac{1}{\csc\theta}=\frac{1}{\frac{1}{\sin\theta}}=1*\frac{\sin\theta}{1}=\sin\theta$

Soh_cah_toa

For the above triangle, what is $\sec (\theta)$ if , and ?

Explanation

Secant is the reciprocal of cosine.

$\sec \left ( \theta\right ) = \frac{1}{\cos \left ( \theta\right )}$

It's formula is:

$\sec(\theta) = \frac{\textup{hypotenuse}}{\textup{adjacent}}$

Substituting the values from the problem we get,

$\sec(\theta) = \frac{17}{15} = 1.13$

Soh_cah_toa

For the above triangle, what is $\cot (\theta)$ if , and ?

Explanation

Cotangent is the reciprocal of tangent.

$\cot \left ( \theta\right ) = \frac{1}{\tan \left ( \theta\right )}$

It's formula is:

$\cot(\theta) = \frac{\textup{adjacent}}{\textup{opposite}}$

Substituting the values from the problem we get,

$\cot(\theta) = \frac{24}{18} = 1.33$

Evaluate:

$2-\sqrt2$

$2-\frac{\sqrt2}{2}$

$2+\frac{\sqrt2}{2}$

$2+\sqrt2$

$2\sqrt2+2$

Explanation

Evaluate each term separately.

$sec(60)= \frac{1}{cos(60)}=\frac{1}{0.5}= 2$

$csc(45)= \frac{1}{sin(45)}= \frac{1}{\frac{\sqrt2}{2}}= \frac{2}{\sqrt2}\times\frac{\sqrt2}{\sqrt2}= \frac{2\sqrt2}{2}=\sqrt2$

$sec(60)-csc(45)=2-\frac{2\sqrt2}{2}=2-\sqrt2$

Pick the ratio of side lengths that would give sec C.

$\sec C=\frac{CB}{AC}$

$\sec C=\frac{AC}{CB}$

$\sec C=\frac{AB}{AC}$

$\sec C=\frac{AC}{AB}$

None of the other answers.

Explanation

$\sec\theta=\frac{1}{\cos\theta }$

Find the ratio of Cosine and take the reciprocal.

$\cos\theta=\frac{AC}{CB}\rightarrow \sec\theta=\frac{1}{\frac{AC}{CB}}=\mathbf{\frac{CB}{AC}}$

If $\sin x=\frac{28}{53}$ , $\cot x=$

$\frac{45}{28}$

$\frac{45}{53}$

$\frac{28}{45}$

$\frac{53}{28}$

Explanation

The sine of an angle in a right triangle (that is not the right angle) can be found by dividing the length of the side opposite the angle by the length of the hypotenuse of the triangle.

From this, the length of the side opposite the angle is proportional to 28, and the length of the hypotenuse is proportional to 53.

Without loss of generality, we'll assume that the sides are actually of length 28 and 53, respectively.

We'll use the Pythagorean theorem to determine the length of the adjacent side, which we'll refer to as .

$\begin{align*} y&=\sqrt{53^2-28^2}\ &=\sqrt{2809-784}\ &=\sqrt{2025}\ &=45 \end{align*}$

The cotangent of an angle in a right triangle (that is not the right angle) is can be found by dividing the length of the adjacent side by the length of the opposite side.

$\cot x=\frac{45}{28}$

Sec, Csc, Ctan

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