Continuing to Payment will take you to apayment page

Purchasing
SparkNotes PLUS
for a group?

Get Annual Plans at a discount when you buy 2 or more!

Price

$24.99$18.74/subscription + tax

Subtotal $37.48 + tax

Save 25%
on 2-49 accounts

Save 30%
on 50-99 accounts

Want 100 or more?
Contact us
for a customized plan.

Your Plan

Payment Details

Payment Details

Payment Summary

SparkNotes Plus

You'll be billed after your free trial ends.

7-Day Free Trial

Not Applicable

Renews June 4, 2023May 28, 2023

Discounts (applied to next billing)

DUE NOW

US $0.00

SNPLUSROCKS20 | 20%Discount

This is not a valid promo code.

Discount Code(one code per order)

SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Choose Your Plan

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thank You!

You’ve successfully purchased a group discount. Your group members can use the joining link below to redeem their group membership. You'll also receive an email with the link.

No URL

Copy

Members will be prompted to log in or create an account to redeem their group membership.

Thanks for creating a SparkNotes account! Continue to start your free trial.

Please wait while we process your payment

Your PLUS subscription has expired

We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.

The ability to solve right triangles has many applications in the real world.
Many of these applications have to do with two-dimensional motion, while others
concern stationary objects. We'll discuss both.

Two-Dimensional Motion

Two-dimensional motion can be represented by a
vector. Every vector can be resolved into a
vertical and a horizontal component. When a vector is combined with its
vertical and horizontal component, a right triangle is formed.

Often the motion of a vehicle of some kind is modeled using a vector. With
limited information, using right triangle solving techniques, it is possible to
find out a lot about the motion of an object in a two-dimensional plane. For
example, if a boat goes 12 miles in a direction 31^{o} north of east, how
far east did it travel? If the boat began at the origin, the problem looks like
this in the coordinate plane:

c = 12 and A = 31^{o}. Then b = c cos(A) 10.29. So the boat went
slightly more than 10 miles east on its journey.

The motion of a projectile in the air can also be easily modeled using a right
triangle. The most common example of this is an airplane's motion. For
example, if an airplane takes off at an angle of elevation of 15^{o} and
flies in a straight line for 3 miles, how high does it get? 3 sin(15) .78. The plane climbs about .78 miles. These types of problems use the
terms angle of elevation and angle of depression, which refer to the angles
created by an object's line of motion and the ground. They can be
mathematically represented by a vector and a horizontal line, usually the x-
axis.

A zero degree angle of elevation or depression means that the object is moving
along the ground--it is not in the air at all. A 90 degree angle of elevation
is motion directly upward, whereas a 90 degree angle of depression is motion
directly downward.

Stationary Objects

Stationary objects that form right triangles can also be examined and understood
by using right triangle solving techniques. One of the most common examples of
a right triangle seen in real life is a situation in which a shadow is cast by a
tall object. For example, if a 40 ft. tree casts a 20 ft. shadow, at what angle
from vertical is the sun shining?

As the picture shows, tan(x) = = . So x = arctan() 26.6^{o}.

Whenever you use a right triangle to model a real-life situation, it is
immensely helpful to draw a picture or diagram of the situation. Then labeling
the parts of the right triangle is easy and the problem can be simply solved.