Reference Angles in Geometry
A reference angle is an angle that is created when two line segments intersect. The line segments can be either lines or arcs, but they must intersect at a point. The point where the line segments intersect is called the vertex, and the reference angle is the angle formed between the line segments at the vertex.
The reference angle is important because it can be used to find the measure of angles that are larger than the reference angle. For example, if you know the measure of a reference angle, you can use that information to find the measure of the angles that are twice or three times the size of the reference angle.
To find the measure of a reference angle, you will need to use basic trigonometric functions. These functions include sine, cosine, and tangent. These functions are used to find angles in right triangles, which are triangle with one 90 degree angle. You can use these functions to find angles that are bigger than 90 degrees by using the symmetries of triangles.
How to Find a Reference Angle
There are a few steps that you need to follow in order to find a reference angle. First, you need to draw a line segment from the vertex to one of the sides of the triangle. This line segment will create two new angles, and one of those angles will be your reference angle.
Next, you need to label your angles. The easiest way to do this is by using letters. The side that you drew your line segment from will be Side a, and the other two sides will be b and c. The two new angles created by your line segment will be Angle A and Angle B. Angle A will be your reference angle.
Now, you need to decide which trigonometric function you will use in order to solve for your reference angle. The function that you choose should be based on what information you know about the triangle. If you know Side b and Side c, then you should use SOHCAHTOA to find Angle A.
SOHCAHTOA stands for:
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
hypotenuse= sqrt(b^2 + c^2) #this finds the hypotenuse using Pythagorean theorem#
Once you have chosen your function, plug in what you know and solve for Angle A. Remember, Angle A is your reference angle! Now that you know how to find a reference angle, let’s look at an example so that you can see how it’s done in practice.
Example: Find Angle A in the triangle below:
b=5 c=12 a=13 #plug in known values for each side# 5²+12²=169 #calculate value of hypotenuse# √169=13 #square root both sides# 13=13 #calculate value of hypotenuse#Angle B=51°90°-51°=39° #subtract 51 from both sides#Angle A=360°-39°=321° #subtract 39 from both sides#Angle C=180°-(321°+51°)=8° #subtract 321 and 51 from both side#Your final answers should be:A= 321°B= 51°C= 8°Note: You could also have solved for Angle C first and then subtracted that value from 180° in order to find Angle A! Practice finding reference angles on your own so that you can become comfortable with using this method. Once you have mastered finding reference angles, try solving problems that involve finding angles that are bigger than your reference angle!
In conclusion, a reference angle is an important concept in geometry because it can help solve complex problems involving larger angles. To review, here are the steps for finding a reference point:[1] Draw a line segment from the vertex.[2] Label your sides and angles.[3] Choose which trigonometric function to use based on what information you have about the triangle.[4] Plug in what you know and solve for the reference angle! Keep practicing so you can become more confident in finding reference angles on your own! As always, if you need additional help or clarification on this concept feel free to reach out to a math teacher or tutor for assistance! Thank you for reading and happy studying!
FAQ
How do you find a reference angle?
To find a reference angle, you will need to use the inverse function of whatever trigonometric function you are using. For example, if you are working with the cosine function, you will need to use the arccosine function. You can usually find these functions on scientific or graphing calculators. The reference angle is the angle between the terminal side of an angle and the x-axis. To find the reference angle, you take the angle (in radians or degrees) and plug it into the inverse function. The output will be the reference angle.
What is a reference angle examples?
A reference angle is the angle between the terminal side of an angle and the x-axis. To find the reference angle, you take the angle (in radians or degrees) and plug it into the inverse function. The output will be the reference angle. For example, if you are working with the cosine function, you will need to use the arc cosine function. You can usually find these functions on scientific or graphing calculators.
What is the reference angle of 90 degrees?
The reference angle of 90 degrees is pi/2 radians. To find the reference angle, you take the angle (in radians or degrees) and plug it into the inverse function. The output will be the reference angle. For example, if you are working with the cosine function, you will need to use the arc cosine function. You can usually find these functions on scientific or graphing calculators.
How do reference angles work?
A reference angle is the angle between the terminal side of an angle and the x-axis. To find the reference angle, you take the angle (in radians or degrees) and plug it into the inverse function. The output will be the reference angle. For example, if you are working with the cosine function, you will need to use the arccosine function. You can usually find these functions on scientific or graphing calculators.
