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SAT II Math I : Trigonometry

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

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Example Question #1 : Trigonometric Identities

Determine the value of: $(1+cot^2(\theta))(1-cos^2(\theta))$

Possible Answers:

$1-cos(\theta)$

$sin(\theta)-cos(\theta)$

$1-csc(\theta)$

Correct answer:

Explanation:

In order to solve $(1+cot^2(\theta))(1-cos^2(\theta))$ , first identify all the recognizable identities.

The identities that can be used for this scenario are:

$sin^2(\theta)+cos^2(\theta)=1$

$1+cot^2(\theta)=csc^2(\theta)$

The first identity can also be manipulated.

$1-cos^2(\theta)=sin^2(\theta)$

Replace $(1+cot^2(\theta))(1-cos^2(\theta))$ with the correct identities and simplify.

$csc^2(\theta)\times sin^2(\theta)= \frac{1}{sin^2(\theta)}\times sin^2(\theta) =1$

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Example Question #42 : Trigonometry

Which of the following is equivalent to: $\frac{1}{1 + cos \ x}{}$ ?

Possible Answers:

sec^2 x + tan^2 x

$cos \ x \ csc^2 x - sin \ x \ sec^2 x$

$csc^2x - cot \ x \ csc \ x$

$sec \ x + tan \ x$

cot^2x - tan^2x

Correct answer:

$csc^2x - cot \ x \ csc \ x$

Explanation:

We can multiply the bottom with its conjugate and obtain:

$\frac{1}{1 + cos \ x} * \frac{1- cos \ x}{1 - cos \ x} = \frac{1 - cos x}{1 - cos^2 x}$

Then we can use the pythagorean identity for the cosines and sines:

$\frac{1 - cos x}{1 - cos^2 x} = \frac{1- cos x}{sin^2 x}$

Finally, we can split the fractions up and translate them into the trigonometric identity:

$= \frac{1}{sin^2x} - \frac{cos \ x}{sin^2x} = (\frac{1}{sin \ x})^2 - \frac{cos \ x}{sin \ x}*\frac{1}{sin \ x}$

$\mathbf{= csc^2 x - cot \ x \ csc \ x}$

Alternatively, you could take this and other answer choices and work the opposite way by translating all of the trigonometric ratios into sines and cosines, using the identities.

Another potential option is to try certain angle values for the original identity and selecting the choice that most closely matches the answer choices when the same angle is substituted (although one must beware of where the function(s) are undefined).

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Example Question #41 : Trigonometry

Simplify sin^2(x) using the trigonometric power reducing formula.

Possible Answers:

$\frac{1-cos(2x)}{1+cos(2x)}$

$\frac{1+cos(2x)}{2}$

$\frac{1+sin(2x)}{2}$

$\frac{1-sin(2x)}{2}$

$\frac{1-cos(2x)}{2}$

Correct answer:

$\frac{1-cos(2x)}{2}$

Explanation:

The power-reducing formulas state that:

$sin^2(x)=\frac{1-cos(2x)}{2}$

$cos^2(x)=\frac{1+cos(2x)}{2}$

$tan^2(x)=\frac{1-cos(2x)}{1+cos(2x)}$

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SAT II Math I : Trigonometry

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

All SAT II Math I Resources

Example Questions

Example Question #1 : Trigonometric Identities

Example Question #42 : Trigonometry

Example Question #41 : Trigonometry

All SAT II Math I Resources

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