Trigonometry : Apply Basic and Definitional Identities

Example Questions

Example Question #1 : Apply Basic And Definitional Identities

Which of the following trigonometric identities is INCORRECT?

Explanation:

Cosine and sine are not reciprocal functions.

and

Example Question #2 : Apply Basic And Definitional Identities

Using the trigonometric identities prove whether the following is valid:

Only in the range of:

Uncertain

Only in the range of:

False

True

True

Explanation:

We begin with the left hand side of the equation and utilize basic trigonometric identities, beginning with converting the inverse functions to their corresponding base functions:

Next we rewrite the fractional division in order to simplify the equation:

In fractional division we multiply by the reciprocal as follows:

If we reduce the fraction using basic identities we see that the equivalence is proven:

Example Question #3 : Apply Basic And Definitional Identities

Which of the following identities is incorrect?

Explanation:

The true identity is  because cosine is an even function.

Example Question #4 : Apply Basic And Definitional Identities

State  in terms of sine and cosine.

Explanation:

The definition of tangent is sine divided by cosine.

Example Question #5 : Apply Basic And Definitional Identities

Simplify.

Explanation:

Using these basic identities:

we find the original expression to be

which simplifies to

.

Further simplifying:

The cosines cancel, giving us

Example Question #6 : Apply Basic And Definitional Identities

Which of the following is the best answer for ?

Explanation:

Write the Pythagorean identity.

Substract  from both sides.

Example Question #7 : Apply Basic And Definitional Identities

Express  in terms of only sines and cosines.

Explanation:

The correct answer is . Begin by substituting , and . This gives us:

.

Example Question #8 : Apply Basic And Definitional Identities

Express  in terms of only sines and cosines.