You are probably familiar with talking about the measure of angles using degrees. Radians are another unit of measure for angles. One radian is the angle swept out by a circular arc that is the same length as its radius. A circle spans or radians. An angle in degrees is typically represented as theta. An angle in radians is normally represented as x.
The conversion from degrees to radians is: which reduces to
To understand the representation of degrees to radians, a unit circle is represented below:
Exercises: | |
Convert to radians: | Convert to degrees: |
Solutions:
The Unit Circle is a circle of radius 1 centered at the origin. Every point on the unit circle corresponds to an angle ?, formed by a line drawn from the origin to the point (?, ?), and the positive x-axis. Notice that drawing a vertical line down to the x-axis from the point (x,y) on the unit circle forms a right triangle with leg lengths x and y, and hypotenuse r =1.
Angles in the Unit Circle:
Now we need a way to relate the coordinates of the coordinates of the point to the angle of the triangle made. We will use the functions sine, cosine, and tangent as shown below.
Another way to think of these trigonometric functions is to name the sides based on their location from the angle ? in the right triangle (see image above). They become:
It may be helpful to remember this by memorizing the acronym SOH-CAH-TOA. These identities can be used to find sines and cosines of angles that are part of a right triangle.
Exercises
Use right-triangle trigonometry (SOH-CAH-TOA) to answer the following.
Problem 1:
Problem 2:
Problem 3:
Solutions:
Problem 1: A
Problem 2: D
Problem 3: B
Special Angles:
The circle is divided up into 4 regions called quadrants. In quadrant I both x and y are positive. In quadrant II x is negative and y is positive. In quadrant III both x and y are negative. In quadrant IV x is positive and y is negative. Since sin, cos, and tan involve x and y coordinates these coordinates retain their positive or negative sign. Note that the radius, r, is always positive.
is positive in the I and II quadrants and negative in the III and IV quadrants.
is positive in the I and IV quadrants and negative in the II and III quadrants.
is positive in the I and III quadrants and negative in the II and IV quadrants.
Exercises:
Find .
Exercises:
Fill out the table with the appropriate reference angle.
Angle | Reference Angle |
Angle | Reference Angle |
Exercises:
Problem 1: Find the reference angle associated with and for .
Problem 2: Find the reference angle associated with and solve for .
Problem 3: Find the reference angle associated with and solve for .
Solutions
Problem 1:
Step 1: Draw a picture.
Step 2:
The angle is in quadrant II so reference angle is . Recall CAH and as you can see in the picture above this would make . In quadrant II, cos is negative so .
Problem 2:
Step 1: Draw a picture.
Step 2:
The angle is in quadrant III so the reference angle is . Recall SOH and as you can see in the picture above this would make . In quadrant III sin is negative so .
Problem 3
Draw a picture.
Problem3-Step 2:
The angle is in quadrant IV so the reference is angle . Recall TOA and as you can see in the picture above this would make . In quadrant IV tan is negative so .
Exercises
Use your knowledge of special angles and reference angles we covered earlier with the new material on special triangles and to fill out this table (Note: Some teachers may want you to be able to fill this chart out from memory on a quiz or test.)
Angle | sin Î¸ | cos Î¸ | tan Î¸ |
---|---|---|---|
Angle | sin Î¸ | cos Î¸ | tan Î¸ |
---|---|---|---|
0 | 1 | 0 | |
1 | |||
1 | 0 | undefined | |
-1 | |||
0 | -1 | 0 | |
1 | |||
-1 | 0 | Undefined | |
-1 | |||
0 | 1 | 0 |
In addition to the three main trigonometric functions there are three reciprocal trigonometric functions. They are the secant, cosecant, and cotangent functions. They have the same sign rules as their normal trigonometric functions based on the quadrant the angle is in. THey also have the same reference angle rules.
Angle | csc Î¸ | sec Î¸ | cot Î¸ |
---|---|---|---|
Angle | csc Î¸ | sec Î¸ | cot Î¸ |
---|---|---|---|
Undefined | 1 | Undefined | |
2 | |||
1 | |||
2 | |||
1 | Undefined | 0 | |
-2 | |||
-1 | |||
2 | |||
Undefined | -1 | Undefined | |
-2 | |||
1 | |||
-2 | |||
-1 | Undefined | 0 | |
2 | |||
-1 | |||
-2 | |||
Undefined | 1 | Undefined |
Let’s take a moment to graph sin, cos, tan, csc, sec, and cot as functions of angle. Put your graphing calculator in radian mode, and use windows y:[-2,2] x:[-5,5].
Solutions:
You may have noticed that these functions repeated themselves, and some of them look very similar. Sine and cosine graphs are actually the same graph, shifted out of phase by . The graphs are periodic, meaning that they repeat every radians. As would be expected, cosecant and secant have the same behavior. Note that sine and cosine have a range of [-1],1] while secant and cosecant have a range of and tangent and cotangent have a range from but have a restricted domain.
(1)
(2)
(3)
(4)
Notice that drawing a vertical line down to the x-axis from the point (x,y) on the unit circle forms a right triangle with leg lengths x and y, and hypotenuse r. The angle created is ?. Because the radius of the unit circle is 1, we can use right-angle trigonometry to determine that ? = cos(Î¸), and ? = sin(?). You should know how to represent sine and cosine on the
unit circle, as shown below.
The inverse trigonometric functions are used to find the angle given a value
They are either represented as:
Or to avoid confusion they can also be represented as:
There is one subtlety to using these functions. Because the trigonometric functions are periodic, there are multiple answers to the inverse functions. When you enter them into a calculator, it returns an answer between â€“ ? and ? radians.
Law of sines
For triangles other than right triangles, right-angle trigonometry cannot be used to find the value of sines and cosines. For finding information about these triangles, the Law of Sines can be very useful. The Law of Sines states that the ratio of the sine of an angle to
the length of its opposite side is the same for all angles in a triangle.
Exercise:
Given:
C=
c=3.2 cm
a=6.7 cm
Find A, B, and b
(Use the picture above)
Step 1: Set up the Line of Sines
,
Step 3: solve for missing angle
If you are given a triangle with the measure of an angle and the two non-corresponding sides, the Law of Cosines will allow you to solve for the unknowns. Notice that once again, the side called a is opposite angle A, b is the side opposite angle B, and c is the side opposite angle C.
Note: If there is a right angle in the triangle, we call that angle C . This makes c the hypotenuse and since cos(C)=0, the Law of Cosines simplifies to the Pythagorean Theorem.
Exercise:
Given:
Find a, B and C using the Law of Cosines
(Use picture above.)
Solution:
Step 1: Set up Equation
To find the area of a triangle you would normally use the formula . To use this formula, you must know or calculate the height of the triangle. Heronâ€™s Formula allows us to calculate the area of the triangle using the lengths of all three sides, without finding the height.
We have talked about the Pythagorean Theorem and how it relates to a unit circle, now we are going to look at how it relates to sine and cosine. The Pythagorean Identity is just as important as the theory and is one you should memorize. It is derived from the following equations:
The Pythagorean identity, sin2 ? + cos2 ? = 1, can be used to obtain more useful identities. If you have a problem that involves a tangent or cotangent squared term these two identities will be useful:
Step 1: Start with the Pythagorean Identity. Divide by .
Negative Angle Identities
To find the negative angle identities, graphing the functions is helpful. First plot the normal function on both sides of the y axis and then reflect over the axis to see how they relate. If you have a graphing calculator you can just plot with a negative theta and compare to the normal graph. The graphs are provided for your review below along with the algebraic identities.
The sum and difference identities are derived from the unit circle using geometric principles. They are very useful for finding values of trigonometric functions that you normally would need a calculator for. You can use sines and cosines you already have memorized to calculate other values. For example, can be calculated as .
Solutions:
Problem 1:
Problem 2:
Problem 3:
Product to Sum Identities
Converting from a product to a sum can be a useful shortcut for simplifying expressions.
Now we need a way to go from a sum to a product. We will use the product to sum identities and work backwards to get these new sums to product identities. The first thing we need to fix is that our side with the sum has A-B inside the function rather than just an A or B so we will need to substitute in to get a single variable.
Let and let and substituting into the following equations:
Start with a product sum identity:
Multiplying these last equations by 2 on each side to come up with the final product identities gives us:
You may have noticed that in the double angle identities section, the cosine identity had a couple of other alternate forms that were not steps in the derivation. We did this because they are used in the derivation of the half angle formulas as shown below. Note: we have replaced A with x to avoid confusion in later steps.
Let and we now have an equation for in terms of functions of just A.
Finishing the substitution with 2x=A, this gives us the final equations:
To get a half angle identity for tangent we use the previous two identities we have already found.
These are more obscure identities that you may need at some point for reference and we will list them so that you know where you can find find them and can have an idea of what is out there.
Inverse Identities:
Any true equation that puts the trigonometric functions into different forms is an identity. Some of these are more useful than others. Of the ones listed above, you will probably only need to know or know how to get the Pythagorean Identities. The others are normally given to you in an equation sheet and you just need to know how to use them. When practicing trigonometric identities you will be asked to verify that a given equation is an identity. To do this, you will need to work with one side of the equation only and end up with the other side of the equation. These can be fairly simple or extremely complex. We will work quite a few of these so that you can get as much experience as you would like working with identities.
Hint: Start working with the most complicated side, it normally will make it easier.
Verify the following identities.
Solution:
Step 1: Write everything in terms of sin and cos on one side.
Step 2: Rearrange equation.
Step 3: Reduce if possible.
Step 4: If both sides of the equation match, it is an identity.
Solution:
Step 1: Write everything in terms of sin and cos on one side.
Step 2: Rearrange equation.
Step 3: Reduce if possible.
Step 4: Apple known identities and rearrange to suite equation
Step 5: If both sides of the equation match, it is an identity.
Solution:
Step 1: Write everything in terms of sin and cos on one side
Step 2: Reduce if possible.
Step 3: Apply known identities
Step 4: If both sides of the equation match, it is an identity.