Download Right Triangle Trigonometry Special Right Triangles

Analytical Geometry
Right Triangle Trigonometry
Right Triangle Trigonometry
What is the shortest distance from point A to point B? What
size ladder do you need to reach the roof of your house? How
high up is that plane? What is the angle of elevation for a
ramp? Each of these questions can be answered using a right
triangle. Having just two measurements, sides or angles, you
can answer any question about a right triangle. All right
triangles share common ratios which make it possible to use
this one simple shape to solve numerous mathematical
problems.
Essential Questions
Why are trigonometric ratios important?
How can we solve problems using right triangles?
Module Minute
Trigonometry is the study of the ratios of sides
and angles in a right triangle. The trigonometry
ratios are sine, cosine, and tangent. Sine is the ratio of the side opposite an acute angle to the hypotenuse of
a right triangle. Cosine is the ratio of the side adjacent to an acute angle to the hypotenuse of a right triangle.
Since these are ratios, similar triangles will have the same ratios. If you know two measurements of a right triangle, you can
use these ratios to find the missing measurements. Using trig ratios and what we know about right triangles, it is possible to
take just two measurements and find each of the other four for a total of three angles and three sides.
Key Words
Adjacent Side ­ The non­hypotenuse side of a triangle that makes an acute angle.
Complementary Angles ­Two angles whose sum is 90°.
Opposite Side ­ In a right triangle, the side of the triangle opposite the vertex of an acute angle, relative to that acute
angle.
Similar Triangles ­ Triangles that have the same shape but not necessarily the same size. A handout of these key words and definitions is also available in the sidebar.
What To Expect
Special Right Triangles Task
Sine vs. Cosine Discussion
Special Right Triangles Quiz
Right Triangle Trig Quiz
Right Triangle Trigonometry Test
Horizons Project
To view the standards from this unit, please download the handout from the sidebar.
Special Right Triangles
Pythagorean Theorem is very handy when we are trying to find missing sides of a right triangle. However, you will find that
once we recognize patterns and ratios, there are faster ways to solve for missing parts of a right triangle.
Watch the presentation below to see some examples of special right triangles.
So, what did you discover from watching the presentation? Let's see if you can remember the special trick!
This is known as a special right triangle. It is called the 45­45­90 triangle. It is named for its angles and because there is a ratio
between the lengths of the sides and the hypotenuse. The hypotenuse will always be always be the same.
times the leg. The two legs will
Example 4:
Given that the length of the hypotenuse of a 45­45­90 triangle is 5, find the legs. Sketch the triangle.
Since we know that the hypotenuse is times the leg, and we were given the hypotenuse is 5, we can divide our given by in order to find the length of the leg.
Example 5:
Now, try a triangle with hypotenuse length 7.
Another Special Right Triangle...
There is another special right triangle.
Let's take a look at one of them.
We know that it has a 90 degree angle where we drew the line of symmetry. We also know that one angle is 60 degrees. That
means the other angle, the one that was bisected is 30 degrees.
We call this the 30­60­90 triangle.
If the hypotenuse of this triangle is 8, what will the short leg of the triangle be?
ANSWER
We know this because, originally, this was an equilateral triangle and the short leg is half of the size of the sides of the original
triangle. In other words, the short leg is half of the hypotenuse.
If we use Pythagorean theorem, we will find that the long leg is .
Notice anything? Try other lengths for the hypotenuse...the ratios will be the same. The short leg is always half of the
hypotenuse and the long leg is always times the short leg.
To Summarize
Special Right Triangles Task
It is now time to complete the Special Right Triangles Task. Download and complete the Right Triangles Task from
the sidebar and submit your completed assignment when finished.
Sine and Cosine
Trigonometry is branch of mathematics that deals with the relationships between the sides and angles of triangles.
We will be learning two trigonometric ratios: sine and cosine.
Trigonometric ratios are ratios of the lengths of two sides in a right triangle.
When solving right triangles we use key words such as adjacent and opposite sides to refer to certain sides in reference to an
angle.
The opposite side will be whatever side is opposite the angle that we are referring to at the time.
The adjacent side will be the side next to the angle which is not the opposite side.
The adjacent and opposite sides will change depending on which angle we are referring to.
The hypotenuse will always be the same. It is the longest side, the side across from the right angle.
Sine
Sine of an angle is equal to the angles opposite side divided by the hypotenuse.
Knowing how to calculate trig ratios such as sine can be helpful in finding missing measurements in real life. For example, look
at the right triangle in the following picture.
With just one more piece of information we can answer many questions about the triangle created by this mountain.
Cosine Another trig function that we can use to solve a right triangle is cosine, written as cos. Cosine of an angle is equal to the ratio
of the adjacent side divided by the hypotenuse.
Sine vs. Cosine Discussion
What is the relationship between the sin of angle and cos of the other acute angle in the same triangle?
It is now time to complete the "Sine vs. Cosine" discussion. A rubric for the discussion is located in the sidebar.
Special Right Triangle Quiz
It is now time to complete the "Special Right Triangle Quiz". You will have a limited amount of time to complete your
quiz; please plan accordingly.
Solving Right Triangles
Example 1:
Set up the trig ratio based on the angle given.
Looking at angle x we can see that we are told the adjacent side and the hypotenuse.
What trig function uses adjacent and hypotenuse?
ANSWER
To solve a right triangle we must find all of the missing sides and angles. We can use
sine, cosine, and the Pythagorean Theorem to find missing sides. We will use inverse
trigonometric ratios to find the measure of the angles.
The inverse trig functions are:
sin­1 and cos­1
Think of inverse trig functions as a way to "undo" sine or cosine in order to find the
measure of the angle.
Example 2:
Given: sin(A) = Find: measure of angle A.
ANSWER Example 3:
Given: cos(B) = .1782
Find: measure of angle B.
ANSWER
Example 4:
Find the measure of angle B.
Remember, the angles of a triangle add up to 180˚.
ANSWER
Example 5:
Using the same triangle from example 4, find the length of AB.
Using angle A, we know the opposite side and we need to find the hypotenuse. We can use the sine ratio.
sin(32) = Next, cross multiply.
sin(32) •(hyp)= 5
Divide both sides by sin(32).
hyp = Finally, type into your calculator.
The hypotenuse is approximately 9.43.
Right Triangles Trig Quiz
It is now time to complete the "Right Triangles Trig Quiz". You will have a limited amount of time to complete your
quiz; please plan accordingly.
Module Wrap Up
Assignment Checklist
In this module you were responsible for completing the following assignments.
Special Right Triangles Task
Sine vs. Cosine Discussion
Special Right Triangle Quiz
Right Triangles Trig Quiz
Right Triangle Trigonometry Test
Horizons Project
Review
Now that you have completed the initial assessments for this module, review the lesson material with the practice activities and
extra resources. Then, continue to the next page for your final assessment instructions.
Standardized Test Preparation
The following problems will allow you to apply what you have learned in this module to how you may see questions
asked on a standardized test. Please follow the directions closely. Remember that you may have to use prior
knowledge from previous units in order to answer the question correctly. If you have any questions or concerns, please
contact your instructor.
Final Assessments
Right Triangle Trigonometry Test
It is now time to complete the "Right Triangle Trigonometry" Test. Once you have completed all self­assessments,
assignments, and the review items and feel confident in your understanding of this material, you may begin. You will
have a limited amount of time to complete your test and once you begin, you will not be allowed to restart your test. Please plan accordingly.
Horizons Project
It is now time to complete the Horizons Project. Download and complete the Horizons Project from the sidebar and
submit your completed assignment when finished. A rubric for this project is located in the sidebar.