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Trigonometry : Sec, Csc, Ctan

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Example Questions

Example Question #1 : Sec, Csc, Ctan

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

Triangle

What is the secant of $\angle A$ ?

Possible Answers:

$\frac{25}{24}$

24/25

7/25

24/7

25/7

Correct answer:

$\frac{25}{24}$

Explanation:

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

Therefore:

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

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Example Question #2 : Sec, Csc, Ctan

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

Possible Answers:

$\cos\theta$

$\sec\theta$

$\cot\theta$

$\tan\theta\sin\theta$

$\sin\theta$

Correct answer:

$\sin\theta$

Explanation:

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

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

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Example Question #3 : Sec, Csc, Ctan

Soh_cah_toa

For the above triangle, what is $\sec (\theta)$ if o = 8 , a = 15 and h = 17 ?

Possible Answers:

0.47

2.13

0.88

0.53

{}1.13

Correct answer:

{}1.13

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,

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

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Example Question #1 : Sec, Csc, Ctan

Soh_cah_toa

For the above triangle, what is $\cot (\theta)$ if o = 18 , a = 24 and h = 30 ?

Possible Answers:

0.75

0.60

1.67

1.25

1.33

Correct answer:

1.33

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,

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

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Example Question #1 : Secant, Cosecant, Cotangent

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

Possible Answers:

$\infty$

$\frac{1}{2}$

$\frac{\sqrt2}{2}$

Correct answer:

Explanation:

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

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

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Example Question #3 : Sec, Csc, Ctan

Evaluate: sec(60)-csc(45)

Possible Answers:

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

$2-\sqrt2$

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

$2\sqrt2+2$

$2+\sqrt2$

Correct answer:

$2-\sqrt2$

Explanation:

Evaluate each term separately.

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

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

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

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Example Question #1 : Sec, Csc, Ctan

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

Possible Answers:

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

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

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

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

None of the other answers.

Correct answer:

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

Explanation:

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

Find the ratio of Cosine and take the reciprocal.

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

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Example Question #7 : Sec, Csc, Ctan

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

Possible Answers:

$\frac{28}{45}$

$\frac{45}{28}$

$\frac{53}{28}$

$\frac{45}{53}$

Correct answer:

$\frac{45}{28}$

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 .

y^2+28^2=53^2

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

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.

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

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Trigonometry : Sec, Csc, Ctan

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Sec, Csc, Ctan

Example Question #2 : Sec, Csc, Ctan

Example Question #3 : Sec, Csc, Ctan

Example Question #1 : Sec, Csc, Ctan

Example Question #1 : Secant, Cosecant, Cotangent

Example Question #3 : Sec, Csc, Ctan

Example Question #1 : Sec, Csc, Ctan

Example Question #7 : Sec, Csc, Ctan

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