We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Trigonometry : Identities with Angle Sums

Study concepts, example questions & explanations for Trigonometry

Create An Account

All Trigonometry Resources

6 Diagnostic Tests 155 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Identities With Angle Sums

Given $\phi = \pi$ , what is $\tan\left ( \theta + \phi \right )$ ?

Possible Answers:

$\tan\left ( \theta\right )$

$\cos\left ( \theta\right )$

$\sin\left ( \theta\right )$

Correct answer:

$\tan\left ( \theta\right )$

Explanation:

We need to use the formula

$\tan\left ( \theta \pm \phi\right ) = \frac{\tan(\theta) \pm \tan(\phi)}{1 \mp \tan(\theta)\tan(\phi)}$

Substituting $\phi = \pi$ , and $\tan (\pi) = 0$ ,

$\tan\left ( \theta + \pi\right ) = \frac{\tan(\theta) + \tan(\phi)}{1 - \tan(\theta)\tan(\phi)} = \frac{\tan(\theta) + 0}{1 - \tan(\theta)\left ( 0\right )} = \tan(\theta)$

Report an Error

Example Question #1 : Identities With Angle Sums

Find the exact value of $\cos 105^{\circ}$ using $105^{\circ} = 60^{\circ}+45^{\circ}$ .

Possible Answers:

$\frac{-\sqrt{2}}{2}$

$\frac{\sqrt{2}+\sqrt{6}}{4}$

$\frac{-(\sqrt{2}+\sqrt{6})}{4}$

$\frac{\sqrt{2}-\sqrt{6}}{4}$

$\frac{-\sqrt{3}}{2}$

Correct answer:

$\frac{\sqrt{2}-\sqrt{6}}{4}$

Explanation:

Our basic sum formula for cosine is:

$\cos(a+b)=\cos a\cos b - \sin a\sin b$

Substituting the relevant angles gives us:

$\cos 105^{\circ}=\cos 60^{\circ}\cos 45^{\circ} - \sin 60^{\circ}\sin 45^{\circ}$

Now substitute in the exact values for each function, simplifying to keep radicals out of the denominator:

$\cos 105^{\circ} = \frac{1}{2}\cdot \frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2}\cdot \frac{\sqrt {2}}{2}$

Multiply and subtract to obtain:

$\frac{\sqrt{2}-\sqrt{6}}{4}$

Report an Error

Example Question #2 : Identities With Angle Sums

Find the exact value of the expression:

$\sin 60^{\circ}\cos 30^{\circ} + \cos 60^{\circ} \sin 30^{\circ}$

Possible Answers:

$\frac{1}{2}$

$\frac{\sqrt{2}}{2}$

The expression is undefined.

Correct answer:

Explanation:

There are two ways to solve this problem. If one recognizes the identity

$\sin (a+b)= \sin a\cos b + \cos a\sin b$ ,

the answer is as simple as:

$\sin 60^{\circ}\cos 30^{\circ} + \cos 60^{\circ}\sin 30^{\circ} = \sin (60^{\circ}+30^{\circ}) = \sin 90^{\circ} = 1$

If one misses the identity, or wishes to be more thorough, you can simplify:

$\sin 60^{\circ}\cos 30^{\circ} + \cos 60^{\circ}\sin 30^{\circ}$

$=\bigg (\bigg(\frac{\sqrt{3}}{2}\bigg)\cdot\bigg(\frac{\sqrt{3}}{2}\bigg)\bigg)+\bigg(\bigg(\frac{1}{2}\bigg)\cdot\bigg(\frac{1}{2}\bigg)\bigg)=\frac{3}{4} + \frac{1}{4}=1$

Report an Error

Example Question #3 : Identities With Angle Sums

Find the exact value of the expression:

$\cos\bigg(\frac{5\pi }{6}\bigg)\cos\bigg(\frac{2\pi}{3}\bigg)+\sin\bigg(\frac{5\pi}{6}\bigg)\sin\bigg(\frac{2\pi}{3}\bigg)$

Possible Answers:

$\frac{\sqrt{3}}{2}$

$-\frac{\sqrt{2}}{2}$

$-\frac{\sqrt{3}}{2}$

$\frac{\sqrt{2}}{2}$

Correct answer:

$\frac{\sqrt{3}}{2}$

Explanation:

The formula for the cosine of the difference of two angles is

$\cos (a-b) = \cos a\cos b + \sin a\sin b$

Substituting, we find that

$a= \frac{5\pi}{6}$

and

$b= \frac{2\pi}{3}$

Therefore, what we are really looking for is

$\cos (a-b)= \cos \bigg(\frac{5\pi}{6}-\frac{2\pi}{3}\bigg) = \cos\bigg(\frac{\pi}{6}\bigg) = \frac{\sqrt{3}}{2}$

Thus,

$\cos \bigg(\frac{5\pi}{6}\bigg)\cos\bigg(\frac{2\pi}{3}\bigg) + \sin\bigg (\frac{5\pi}{6}\bigg)\sin\bigg(\frac{2\pi}{3}\bigg) = \frac{\sqrt{3}}{2}$

Report an Error

Example Question #34 : Trigonometric Identities

Find the exact value of $\tan 75^{\circ}$ using $\tan 30^{\circ}$ and $\tan 45^{\circ}$ .

Possible Answers:

$\frac{3\sqrt{2}}{3}$

$2+\sqrt{3}$

The quantity cannot be found exactly using the given information.

$2-\sqrt{3}$

$\frac{2+\sqrt{3}}{2}$

Correct answer:

$2+\sqrt{3}$

Explanation:

The sum identity for tangent states that

$\tan (a+b) = \frac{\tan a + \tan b}{1-\tan a\tan b}$

Substituting known values for $\tan 30^{\circ}$ and $\tan 45^{\circ}$ , we have

$\tan (30^{\circ} + 45^{\circ}) = \frac{1+\frac{1}{\sqrt{3}}}{1-\frac{1}{\sqrt{3}}}$

For ease, multiply all terms by $\sqrt{3}$ to get $\frac{\sqrt{3}+1}{\sqrt{3}-1}$ .

At this point, multiply both halves of the fraction by the conjugate of the denominator:

$\bigg(\frac{\sqrt{3}+1}{\sqrt{3}-1}\bigg)\cdot\bigg(\frac{\sqrt{3}+1}{\sqrt{3}+1}\bigg) = \frac{3+2\sqrt{3}+1}{3-1}$

Finally, simplify.

$\frac{3+2\sqrt{3}+1}{3-1} = \frac{4+2\sqrt{3}}{2} = 2+\sqrt{3}$

So, $\tan 75^{\circ} = 2+\sqrt{3}$ .

Report an Error

Example Question #6 : Identities With Angle Sums

Suppose we have two angles, and , such that:

$sin\ A = 0.10$

$cos\ B = 0.10$

Furthermore, suppose that angle is located in the first quadrant and angle is located in the fourth.

What is the measure of:

Sin(A - 2B)?

Possible Answers:

0.2

-0.10

0.10

0.08

-0.02

Correct answer:

0.10

Explanation:

We can calculate some missing values using the pythagorean identities.

$sin^2\ B = 1 - cos^2B = 1 - 0.01 = 0.99$

$sin\ B = {0.99}^\frac{1}{2} = -0.995$

(Note the negative sign, because is in the fourth quadrant, where the sine of the angle is always negative).

$cos\ A = \sqrt{1 - sin^2A} = \sqrt{1 - 0.01} = \sqrt{0.99} = 0.995$

Note the positive value, since is in the first quadrant, where cosine is positive.

Now using the rules for double angles:

$cos\ 2B = 2\ cos^2 B - 1 = 2 * 0.1^2 - 1 = -0.98$

$sin\ 2B = 2\ cos\ B\ sinB = 2 \ * 0.1 \ * -0.995 = -0.199$

And then the angle subtraction formula:

$sin(A - 2B) = sin\ A \ cos \ 2B\ - cos\ A \ sin 2B$

$= (0.1 * -0.98) \ - (0.995 * -0.199)$

$= \textbf{0.10}$

Report an Error

Example Question #2 : Identities With Angle Sums

Calculate $sin(105^\circ)$ .

Possible Answers:

$\frac{\sqrt{3}+\sqrt{2}}{2}$

$\frac{3\sqrt{6}+1}{4}$

$\frac{\sqrt{6}+\sqrt{2}}{2}$

$\frac{3\sqrt{6}+\sqrt3}{4}$

$\frac{\sqrt{6}+\sqrt{2}}{4}$

Correct answer:

$\frac{\sqrt{6}+\sqrt{2}}{4}$

Explanation:

Recall the formula for the sine of the sum of two angles:

sin(x+y)=sin(x)cos(y)+sin(y)cos(x)

Here, we can evaluate $sin(105^{\circ})$ by noticing that $105^{\circ}=60^{\circ}+45^{\circ}$ and applying the above formula to the sines and cosines of these two angles.

$sin(60^{\circ}+45^{\circ})=sin(60^{\circ})cos(45^{\circ})+sin(45^{\circ})cos(60^{\circ})$

$=(\frac{\sqrt3}{2})(\frac{\sqrt2}{2})+(\frac{\sqrt2}{2})(\frac{1}{2})$

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

Hence,

$sin(105^{\circ})=\frac{\sqrt6+\sqrt2}{4}$

Report an Error

Example Question #34 : Trigonometry

What is the value of $\cos(540)$ , using the sum formula.

Possible Answers:

$\frac{1}{2}$

Correct answer:

Explanation:

The formula for

$\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)$ .

We can expand

540=180+360 ,

where A=180 and B=360 .

Substituting these values into the equation, we get

$\cos(180+360)=\cos(180)\cos(360)-\sin(180)\sin(360)$ .

The final answer is -1, using what we know about the unit circle values.

Report an Error

Example Question #37 : Trigonometry

Simplify the given expression.

$\sin(x+y)-sin(x-y)$

Possible Answers:

$\sin 2xy$

$2\cos x\sin y$

$\cos x\sin y$

$\cos^2 x$

$\sin 2y$

Correct answer:

$2\cos x\sin y$

Explanation:

This problem requires the use of two angle sum/difference identities:

$\sin(x+y)=\sin x\cos y+\cos x\sin y$

$\sin(x-y)=\sin x\cos y-\cos x\sin y$

Using these identities, we get

$(\sin x\cos y+\cos x\sin y)-(\sin x\cos y-\cos x\sin y)$

which simplifies to

$\sin x\cos y+\cos x\sin y-\sin x\cos y+\cos x\sin y$

which equals

$2\cos x\sin y$

Report an Error

Example Question #35 : Trigonometric Identities

Possible Answers:

$\small \frac{1}{2}$

$\small 2$

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

$\small 1$

$\small \frac{1}{3}$

Correct answer:

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

Explanation:

Solution

Report an Error

View Trigonometry Tutors

Shifrah
Certified Tutor

University of Maryland-Baltimore County, Bachelor of Science, Biology, General. Massachusetts College of Pharmacy and Health ...

View Trigonometry Tutors

Mohammad Talha
Certified Tutor

University of Toronto, Mississauga, Bachelor, Chemistry, and Environmental Science.

View Trigonometry Tutors

Joseph
Certified Tutor

Rutgers University-New Brunswick, Bachelor in Arts, Chemistry. Rutgers Graduate School of Biomedical Sciences, Current Grad S...

All Trigonometry Resources

6 Diagnostic Tests 155 Practice Tests Question of the Day Flashcards Learn by Concept

Trigonometry Tutors in Top Cities:

Popular Courses & Classes

LSAT Courses & Classes in Miami, SSAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Boston, ACT Courses & Classes in Atlanta, SAT Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in San Diego, SSAT Courses & Classes in Philadelphia, SSAT Courses & Classes in Atlanta, SSAT Courses & Classes in Houston, GMAT Courses & Classes in Philadelphia

Popular Test Prep

SAT Test Prep in Washington DC, MCAT Test Prep in Philadelphia, GRE Test Prep in San Diego, SSAT Test Prep in New York City, GMAT Test Prep in Philadelphia, SSAT Test Prep in Denver, ISEE Test Prep in Los Angeles, GRE Test Prep in Boston, GRE Test Prep in Washington DC, SSAT Test Prep in Seattle

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Trigonometry : Identities with Angle Sums

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Identities With Angle Sums

Example Question #1 : Identities With Angle Sums

Example Question #2 : Identities With Angle Sums

Example Question #3 : Identities With Angle Sums

Example Question #34 : Trigonometric Identities

Example Question #6 : Identities With Angle Sums

Example Question #2 : Identities With Angle Sums

Example Question #34 : Trigonometry

Example Question #37 : Trigonometry

Example Question #35 : Trigonometric Identities

All Trigonometry Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: