Trigonometry : Identities of Halved Angles

Study concepts, example questions & explanations for Trigonometry

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Example Questions

Example Question #1 : Identities Of Halved Angles

Simplify the function below:

Possible Answers:

Correct answer:

Explanation:

We need to use the following formulas:

a) 

 

b) 

 

c) 

 

d) 

 

e) 

 

We can simplify  as follows:

 

Example Question #2 : Identities Of Halved Angles

Find  if  and .

Possible Answers:

Correct answer:

Explanation:

The double-angle identity for sine is written as

and we know that 

Using , we see that , which gives us 

Since we know  is between  and , sin  is negative, so . Thus,

.

Finally, substituting into our double-angle identity, we get

Example Question #3 : Identities Of Halved Angles

Find the exact value of  using an appropriate half-angle identity.

Possible Answers:

Correct answer:

Explanation:

The half-angle identity for sine is:

If our half-angle is , then our full angle is . Thus,

The exact value of  is expressed as , so we have

Simplify under the outer radical and we get

Now simplify the denominator and get

Since  is in the first quadrant, we know sin is positive. So,

Example Question #1 : Identities Of Halved Angles

Which of the following best represents ?

Possible Answers:

Correct answer:

Explanation:

Write the half angle identity for cosine.

Replace theta with two theta.

Therefore:

Example Question #5 : Identities Of Halved Angles

What is the amplitude of ?

Possible Answers:

Correct answer:

Explanation:

The key here is to use the half-angle identity for to convert it and make it much easier to work with.

In this case, , so therefore...

Consequently, has an amplitude of .

Example Question #6 : Identities Of Halved Angles

If , then calculate .

Possible Answers:

Correct answer:

Explanation:

Because , we can use the half-angle formula for cosines to determine .

In general,

for .

For this problem,

                      

                      

                     

                      

 

Hence, 

 

Example Question #7 : Identities Of Halved Angles

What is ?

Possible Answers:

Correct answer:

Explanation:

Let ; then

.

 

We'll use the half-angle formula to evaluate this expression.

 

Now we'll substitute  for .

 

 is in the first quadrant, so  is positive. So

.

Example Question #8 : Identities Of Halved Angles

What is , given that  and  are well defined values?

Possible Answers:

Correct answer:

Explanation:

Using the half angle formula for tangent, 

,

we plug in 30 for .

We also know from the unit circle that  is  and  is .

Plug all values into the equation, and you will get the correct answer. 

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