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SAT II Math I : Secant, Cosecant, Cotangent

Study concepts, example questions & explanations for SAT II Math I

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Example Questions

Example Question #1 : Secant, Cosecant, Cotangent

If $\cot \alpha = 5$ and $\sin \alpha < 0$ , what is the value of $\sec \alpha$ ?

Possible Answers:

$\sqrt{26}$

$-\frac{1}{\sqrt{26}}$

$-\frac{\sqrt{26}}{5}$

$\frac{\sqrt{26}}{5}$

$-\frac{5}{\sqrt{26}}$

Correct answer:

$-\frac{\sqrt{26}}{5}$

Explanation:

Since cotangent is positive and sine is negative, alpha must be in quadrant III. $\cot \alpha = \frac{x}{y}$ then implies that (x,y)=(-5,-1) is a point on the terminal side of alpha.

$r=\sqrt{(x^2+y^2)}=\sqrt{26}$

$\sec \alpha = \frac{r}{x} = \frac{\sqrt{26}}{-5}$

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Example Question #2 : Secant, Cosecant, Cotangent

If $\csc \theta < 0$ and $\tan \theta > 0$ , then which of the following must be true about $\theta$ .

Possible Answers:

$\frac{\pi}{2} < \theta < \pi$

$0 < \theta < \frac{\pi}{2}$

$-\frac{\pi}{2} < \theta < 0$

$\frac{3\pi}{2} < \theta < 2\pi$

$\pi < \theta < \frac{3\pi}{2}$

Correct answer:

$\pi < \theta < \frac{3\pi}{2}$

Explanation:

Since cosecant is negative, theta must be in quadrant III or IV.

Since tangent is positive, it must be in quadrant I or III.

Therefore, theta must be in quadrant III.

Using a unit circle we can see that quadrant III is when theta is between $\pi$ and $\frac{3\pi}{2}$ .

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Example Question #3 : Secant, Cosecant, Cotangent

The point (12, 5) lies on the terminal side of an angle in standard position. Find the secant of the angle.

Possible Answers:

$\frac{13}{12}$

$\frac{13}{5}$

$\frac{5}{13}$

$\frac{5}{12}$

$\frac{12}{13}$

Correct answer:

$\frac{13}{12}$

Explanation:

Secant is defined to be the ratio of to where is the distance from the origin.

The Pythagoreanr Triple 5, 12, 13 helps us realize that r = 13 .

Since x = 12 , the answer is $\frac{13}{12}$ .

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Example Question #4 : Secant, Cosecant, Cotangent

Given angles and in quadrant I, and given,

$\sin x = \frac{3}{5}$ and $\cos y = \frac {5}{13}$ ,

find the value of $\csc (x+y)$ .

Possible Answers:

$\frac {63}{65}$

$\frac {65}{64}$

$\frac {65}{63}$

$\frac {65}{64}$

$\frac {99}{65}$

Correct answer:

$\frac {65}{63}$

Explanation:

Use the following trigonometric identity to solve this problem.

$\csc (x+y) = \frac {1}{\sin (x+y)} = \frac {1}{\sin x \cos y + \cos x \sin y}$

Using the Pythagorean triple 3,4,5, it is easy to find $\cos x = \frac {4}{5}$ .

Using the Pythagorean triple 5,12,13, it is easy to find $\sin y = \frac {12}{13}$ .

So substituting all four values into the top equation, we get

$\csc (x+y) = \frac {1}{\frac {3}{5} \cdot \frac {5}{13} + \frac {4}{5} \cdot \frac {12}{13}} = \frac {65}{63}$

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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:

$\frac{\sqrt2}{2}$

$\infty$

$\frac{1}{2}$

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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SAT II Math I : Secant, Cosecant, Cotangent

Study concepts, example questions & explanations for SAT II Math I

All SAT II Math I Resources

Example Questions

Example Question #1 : Secant, Cosecant, Cotangent

Example Question #2 : Secant, Cosecant, Cotangent

Example Question #3 : Secant, Cosecant, Cotangent

Example Question #4 : Secant, Cosecant, Cotangent

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

All SAT II Math I Resources

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