Precalculus : Fundamental Trigonometric Identities

Example Questions

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Example Question #1 : Fundamental Trigonometric Identities

Simplify .

Explanation:

Write the Pythagorean Identity.

Reorganize the left side of this equation so that it matches the form:

Subtract cosine squared theta on both sides.

Multiply both sides by 3.

Example Question #1472 : Pre Calculus

Which of the following statements is false?

Explanation:

Of the six trigonometric functions, four are odd, meaning . These four are:

• sin x
• tan x
• cot x
• csc x

That leaves two functions which are even, which means that . These are:

• cos x
• sec x

Of the aforementioned, only  is incorrect, since secant is an even function, which implies that

Example Question #1 : Fundamental Trigonometric Identities

Find the value of .

Explanation:

Rewrite  by odd and even identities.

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

Example Question #2 : Fundamental Trigonometric Identities

Simplify:

Explanation:

Write the even and odd identities for sine and cosine.

Rewrite the expression  and evaluate.

Example Question #1 : Fundamental Trigonometric Identities

Simplify:

Explanation:

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

Example Question #6 : Fundamental Trigonometric Identities

Which of the following is equivalent to the expression:

Explanation:

Which of the following is equivalent to the expression:

Begin by recalling the following identity:

Next, recall the relationship between cotangent and tangent:

As well as the relationship between tangent, sine and cosine

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

Next, we can rewrite our cotangent as tangent

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.

This may look good on the surface, but recall

Example Question #1 : Fundamental Trigonometric Identities

Simplify:

Explanation:

Write the reciprocal identity for cosecant.

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

Example Question #1481 : Pre Calculus

Determine which of the following is equivalent to .

Explanation:

Rewirte  using the reciprocal identity of cosine.

Example Question #1 : Fundamental Trigonometric Identities

Which of the following is similar to

?

Explanation:

Write the reciprocal/ratio identity for cosecant.

Replace cosecant with sine.

Example Question #1 : Fundamental Trigonometric Identities

Evaluate:

Explanation:

Rewrite  in terms of sine and cosine.

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

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.

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