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Precalculus : Fundamental Trigonometric Identities

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Example Question #1 : Fundamental Trigonometric Identities

Simplify $3sin^2\theta-3$ .

Possible Answers:

$-3tan\theta$

$3-3cos^2\theta$

$3cos^2\theta$

$-3cos^2\theta$

Correct answer:

$-3cos^2\theta$

Explanation:

Write the Pythagorean Identity.

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

Reorganize the left side of this equation so that it matches the form: $3sin^2\theta-3$

Subtract cosine squared theta on both sides.

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

Multiply both sides by 3.

$3(sin^2\theta-1=-cos^2\theta)$

$3sin^2\theta-3=-3cos^2\theta$

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Example Question #1 : Fundamental Trigonometric Identities

Which of the following statements is false?

Possible Answers:

sin x + cos x = -sin(-x) + cos (-x)

sec(-x^2) = -sec(x^2)

$sin(x)cos(x)tan(x) = \frac{1}{cot(-x)sec(-x)csc(-x)}$

csc x tan(-2x) = csc (-x) tan 2x

tan x + tan(-x) = 0

Correct answer:

sec(-x^2) = -sec(x^2)

Explanation:

Of the six trigonometric functions, four are odd, meaning f(-x) = -f(x) . These four are:

sin x
tan x
cot x
csc x

That leaves two functions which are even, which means that f(-x) = f(x) . These are:

cos x
sec x

Of the aforementioned, only sec(-x^2) = -sec(-x^2) is incorrect, since secant is an even function, which implies that sec(-x^2) = sec(x^2)

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Example Question #1 : Fundamental Trigonometric Identities

Find the value of $\sin(-15^\circ)$ .

Possible Answers:

$\frac{-\sqrt6 -\sqrt2}{4}$

$\frac{\sqrt6 -\sqrt2}{2}$

$\frac{\sqrt6 -\sqrt2}{4}$

$\frac{\sqrt2+\sqrt6}{4}$

$\frac{\sqrt2 -\sqrt6}{4}$

Correct answer:

$\frac{\sqrt2 -\sqrt6}{4}$

Explanation:

Rewrite sin(-15) by odd and even identities.

sin(-15)=-sin(15)

Use the difference identity of sine, and choose the special angles 45 and 30, since their difference equals to 15.

sin (A -B) = ( sin A )( cos B ) - ( cos A )( sin B )

sin (45 -30) = ( sin 45 )( cos 30) - ( cos45 )( sin 30 )

$sin (45 -30) = (\frac{\sqrt2}{2})(\frac{\sqrt3}{2}) -(\frac{\sqrt2}{2})(\frac{1}{2})$

$sin (15) = \frac{\sqrt6}{4} -\frac{\sqrt2}{4}$

$-sin(15)=-(\frac{\sqrt6}{4} -\frac{\sqrt2}{4})=\frac{\sqrt2}{4} -\frac{\sqrt6}{4}$

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Example Question #4 : Fundamental Trigonometric Identities

Simplify: sin(-30)+cos(-30)

Possible Answers:

$\frac{\sqrt3+1}{2}$

$\frac{\sqrt3-1}{2}$

$\frac{\sqrt3-1}{3}$

$-\frac{1}{2}$

Correct answer:

$\frac{\sqrt3-1}{2}$

Explanation:

Write the even and odd identities for sine and cosine.

sin(-x)=-sin(x)

cos(-x)=cos(x)

Rewrite the expression sin(-30)+cos(-30) and evaluate.

$-sin(30)+cos(30) =-\frac{1}{2}+\frac{\sqrt3}{2}= \frac{\sqrt3-1}{2}$

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Example Question #5 : Fundamental Trigonometric Identities

Simplify: sec(-45)-sec(-45)

Possible Answers:

$\frac{\sqrt2}{2}$

Correct answer:

Explanation:

In order to simplify sec(-45)-sec(-45) , rewrite the expression after applying the rule of odd-even identities for the secant function.

sec(-x)=sec(x)

sec(-45)-sec(-45)=sec(45)-sec(45)=0

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Example Question #6 : Fundamental Trigonometric Identities

Which of the following is equivalent to the expression:

$cot(-\Theta )$

Possible Answers:

$\frac{sin{\Theta}}{cos(-\Theta)}$

$\frac{cos(-\Theta)}{sin(\Theta)}$

$\frac{-cos(\Theta)}{sin(\Theta)}$

$\frac{sin{\Theta}}{cos(\Theta)}$

Correct answer:

$\frac{-cos(\Theta)}{sin(\Theta)}$

Explanation:

Which of the following is equivalent to the expression:

$cot(-\Theta )$

Begin by recalling the following identity:

$cot(-\Theta )=-cot(\Theta)$

Next, recall the relationship between cotangent and tangent:

$cot(\Theta)=\frac{1}{tan(\Theta)}$

As well as the relationship between tangent, sine and cosine

$Tan(\Theta)=\frac{sin{\Theta}}{cos(\Theta)}$

So to put it all together, we can pull out the negative sign from our original expression:

$cot(-\Theta )=-cot(\Theta)$

Next, we can rewrite our cotangent as tangent

$-cot(\Theta)=\frac{-1}{tan(\Theta)}$

Finally, we can change our tangent to sine and cosine, but because we are dealing with the reciprocal of tangent, we will need the reciprocal of our identity.

$-cot(\Theta)=\frac{-1}{tan(\Theta)}=\frac{-cos(\Theta)}{sin(\Theta)}$

Making our answer:

$\frac{-cos(\Theta)}{sin(\Theta)}$

Beware trap answer:

$\frac{cos(-\Theta)}{sin(\Theta)}$

This may look good on the surface, but recall

$cos(-\Theta)\neq -cos(\Theta),cos(-\Theta)=cos(\Theta)$

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Example Question #7 : Fundamental Trigonometric Identities

Simplify:

$\frac{2}{csc(2t)}$

Possible Answers:

$4cos(\theta)sin(\theta)$

$2cos(\theta)-2sin(\theta)$

$\frac{cos(\theta)sin(\theta)}{4}$

$2sin(\theta)$

$4sin(\theta)$

Correct answer:

$4cos(\theta)sin(\theta)$

Explanation:

Write the reciprocal identity for cosecant.

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

Rewrite the expression and use the double angle identities for sine to simplify.

$\\\frac{2}{csc(2t)}\\ \\= \frac{2}{\frac{1}{sin(2\theta)}}\\ \\= 2sin(2\theta) \\ \\= 2(2sin(\theta)cos(\theta))$

$=4cos(\theta)sin(\theta)$

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Example Question #8 : Fundamental Trigonometric Identities

Determine which of the following is equivalent to 2sec(10x) .

Possible Answers:

sec(5x)

sec(20x)

$\frac{2}{cos(10x)}$

2cos(20x)

$\frac{2}{cos(20x)}$

Correct answer:

$\frac{2}{cos(10x)}$

Explanation:

Rewirte 2sec(10x) using the reciprocal identity of cosine.

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

$\\2sec(10x)\\ \\=2\left(\frac{1}{cos(10x)}\right)\\ \\= \frac{2}{cos(10x)}$

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Example Question #9 : Fundamental Trigonometric Identities

Which of the following is similar to

$\frac{3}{csc(\theta)}$ ?

Possible Answers:

$sin(3\theta)$

$3cos(\theta)$

$3sin(\theta)$

$\frac{3}{sin(\theta)}$

$\frac{sin(\theta)}{3}$

Correct answer:

$3sin(\theta)$

Explanation:

Write the reciprocal/ratio identity for cosecant.

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

Replace cosecant with sine.

$\frac{3}{csc(\theta)}=\frac{3}{\frac{1}{sin(\theta)}}= 3sin(\theta)$

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Example Question #10 : Fundamental Trigonometric Identities

Evaluate:

$\frac{sec(\theta)}{tan(\theta)}-\frac{tan(\theta)}{sec(\theta)}$

Possible Answers:

$\frac{cos(\theta)-sin(\theta)}{cos^2(\theta)}$

$\frac{1+sin(\theta)}{sin(\theta)}$

$\frac{1-sin(\theta)}{sin(\theta)}$

$\frac{cos(\theta)+sin(\theta)}{sin(\theta)}$

$\frac{cos^2(\theta)}{sin(\theta)}$

Correct answer:

$\frac{cos^2(\theta)}{sin(\theta)}$

Explanation:

Rewrite $\frac{sec(\theta)}{tan(\theta)}-\frac{tan(\theta)}{sec(\theta)}$ in terms of sine and cosine.

$\frac{sec(\theta)}{tan(\theta)}-\frac{tan(\theta)}{sec(\theta)}= \frac{1}{cos(\theta)}\div\frac{sin(\theta)}{cos(\theta)}-\frac{sin(\theta)}{cos(\theta)}\div \frac{1}{cos(\theta)}$

Dividing fractions is the same as multiplying the numerator by the reciprocal of the denominator.

$\frac{1}{cos(\theta)}\times\frac{cos(\theta)}{sin(\theta)}-\frac{sin(\theta)}{cos(\theta)}\times cos(\theta)=\frac{1}{sin(\theta)}-sin(\theta)$

Multiply the second term by sine to get a common denominator. Then after subtracting the second term from the first you can see that a Pythagorean Identity is in the numerator.

Reducing further we arrive at the final answer.

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

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Precalculus : Fundamental Trigonometric Identities

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Example Questions

Example Question #1 : Fundamental Trigonometric Identities

Example Question #1 : Fundamental Trigonometric Identities

Example Question #1 : Fundamental Trigonometric Identities

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