The goniometric circumference has a radius (r) equal to 1, which implies that its circumference is 2π since every circumference is calculated by multiplying 2πr. This circumference allows you to easily see the relationship between the coordinates of the cosine and the sine, and also the measure of the angles expressed in radians. Once you become familiar with goniometric circumference, you can more easily understand trigonometry, geometry, and calculus. At first it may seem difficult, but in reality it is much simpler than you think. There are some tricks that will help you memorize these concepts easily.
Steps
Method 1 of 3: Learn to Remember Radians with the TRSP Cheat
Step 1. Learn to use TRSP
TRSP stands for "All Subtracts Adds Prime". You can use a mnemonic to help you remember it. These acronyms will help you remember how to find the radians of each angle. Unfortunately, radians vary by quadrant, although they share the same denominators. That's because the radians range from 0 to 2π.
Everything:
you must memorize all the radians of the first quadrant.
Subtraction:
To obtain the numerator of each radian in the second quadrant, you must subtract 1 from the denominator of the angle corresponding to this in the first quadrant.
Sum:
To obtain the numerator of each radian in the third quadrant, you must add 1 to the denominator of the angle corresponding to this in the first quadrant.
Cousins:
Each radian in the fourth quadrant begins with a prime number.
Step 2. Notice that the x-axis is not a fraction
It is better to see the x-axis as a whole number. The positive side is 0 or 2π, while the negative is 1π. This is because the top of the circle itself measures 1π, plus the bottom of the circle which also measures 1π. The negative side of the x-axis is the first half of the circle and the positive side is the beginning and end of the circle.
Step 3. Notice that the denominator of the y-axis is 2
Since the entire top of the circle measures 1π, it makes sense for the measure of the positive side of the y-axis to be 1π / 2. This occurs because the y-axis divides the top of the circle in half. Similarly, the bottom of the circle is 3π / 2 because the y-axis divides it in half.
If you find it difficult to remember that the negative y-axis is 3π / 2, you can find the radians of the third quadrant using the addition trick
Step 4. Observe that all the quadrants share the same denominators:
6, 4 and 3. This will make it easier for you to remember the radians. The number 3 is always close to the y-axis, while the number 6 is always close to the x-axis. It may seem confusing to you, or it may be easier for you to remember that the smaller numbers are at the top or bottom, while the larger numbers are side by side.
- The denominators in quadrant 1 are in this order: 6, 4, 3.
- The denominators in quadrant 2 are in this order: 3, 4, 6.
- The denominators in quadrant 3 are in this order: 6, 4, 3.
- The denominators in quadrant 4 are in this order: 3, 4, 6.
Step 5. Learn all the radians of the angles in the first quadrant
A radian is the measure of an angle. Each measurement is expressed as a function of π since the circumference of every circle is based on this constant. The radians of a goniometric circle range from 0 to 2π. Most of the angles in a circle are a fraction of π. Next, you will see the radian measure of the first quadrant:
- 0 degree angles measure 0.
- 30 degree angles measure π / 6.
- 45 degree angles measure π / 4.
- Angles of 60 degrees measure π / 3.
- 90 degree angles measure π / 2.
Step 6. R Set the denominator 1 to get the numerators of the second quadrant. Now that you know the pattern of denominators explained above, it will be easier for you to remember the measures of all the angles. In the second quadrant, you know that the denominators are 3, 4 and 6. Simply subtract 1 digit from the denominator and you will have the value of the numerator of the fraction. Don't forget to add π in the numerator. Next, you will see the radians of the angles in the second quadrant:
- 120 degree angles measure 2π / 3.
- 135 degree angles measure 3π / 4.
- 150 degree angles measure 5π / 6.
- 180-degree angles measure π (remember, as explained above, this is the negative x-axis).
Step 7. S Add 1 to the denominator to get the numerators of the third quadrant. Remember that the denominators of the third quadrant are 6, 4 and 3. The numerator the measure of each radian will be the denominator + 1 multiplied by π. Next, you'll see the radians of the angles in the third quadrant:
- The 210 degree angles measure 7π / 6.
- 225 degree angles measure 5π / 4.
- 240 degree angles measure 4π / 3.
- 270 degree angles measure 3π / 2, since it is the negative y-axis. Luckily the quadrant trick works at this angle!
Step 8. Use the prime numbers to obtain the numerators of the fourth quadrant
The trick to finding the numerators of the fourth quadrant measurements is as simple as remembering the prime numbers 3, 5, 7, and 11. Next, you'll see the radians of the angles in the fourth quadrant:
- 270 degree angles measure 3π / 2 radians.
- 300-degree angles have a 5 in the numerator and their measure is 5π / 3.
- Angles of 315 degrees have a 7 in the numerator and their measure is 7π / 4.
- 330 degree angles have an 11 in the numerator and their measure is 11π / 6.
- Finally, the circle ends at a 360-degree angle that equates to a radian of 2π (remember that this is the positive x-axis as explained above).
Method 2 of 3: Using the Left Hand Trick for Sine and Cosine
Step 1. Spread the fingers of your left hand so that the thumb and little finger are at right angles
That is the part of the circle where both the x and y coordinates are positive.
The thumb and little finger should be at a right angle. The little finger will be the x-axis and the thumb is the y-axis.
The cosine is the x coordinate of an angle and the sine the y coordinate.
Step 2. Imagine that each finger represents an angle in the first quadrant
As you move to other quadrants, the angle measure will change. However, the coordinates of the sine and the cosine will be the same integer, although it can change from positive to negative. This means that you can use the left hand trick to find the coordinates of any quadrant! This is how you should label your fingers:
- The little finger represents an angle of 0 degrees. The 0 degree angle falls on the x axis. It is the starting point of your circle and therefore it is worth 0.
- The ring finger represents at a 30 degree angle.
- The middle finger represents a 45 degree angle.
- The index finger represents a 60 degree angle.
- The thumb represents a 90 degree angle.
Step 3. Find the cosine coordinate of an angle by counting the fingers that are to the left
Lower the finger that you are going to use to represent the angle for which you want to find the cosine. Count the number of fingers that are to the left of the finger representing the angle. Then take the square root of this number and divide by 2 to find the coordinates.
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For example, if you wanted to find the coordinates of a 30 degree angle, you would lower your ring finger. To the left of that finger is the thumb, index and middle finger, that is, 3 fingers. This means that the cosine coordinate is 32 { displaystyle { frac { sqrt {3}} {2}}}
. Esta es la respuesta final ya que no es posible simplificar más la fracción.
- Si tuvieras que hallar el coseno de un ángulo de 0 grados, deberías bajar el meñique y contar 4 dedos a la izquierda. La ecuación es 42{displaystyle {frac {sqrt {4}}{2}}}
. Como la raíz cuadrada de 4 es 2, entonces 2/2=1. Ese es el coseno.
Step 4. Obtain the coordinate of the sine of an angle by counting the fingers to the right
Lower your finger again and then count the number of fingers to your right. Find the square root of this number and divide by 2.
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In the example above, for a 30 degree angle there is only one finger on the right: the little finger. This means that the sine coordinate is 12 { displaystyle { frac { sqrt {1}} {2}}}
. Como la raíz cuadrada de 1 es 1, puedes escribir simplemente 1/2.
- Para un ángulo de 0 grados, no hay ningún dedo a la derecha del meñique. Esto quiere decir que el seno es 0.
Step 5. Swap the sign of the coordinates to represent other quadrants
Each quadrant has its own positive or negative sign. It is easier to identify when you look at the circle on a grid. The first quadrant is between the positive x and positive x axes, so both coordinates are positive. The second quadrant is between the positive y and negative x axes, therefore it is negative and positive. Observe the signs corresponding to the coordinates of each quadrant:
- The coordinates of quadrant 1 are (+, +).
- Quadrant 2 coordinates are (-, +).
- The coordinates of quadrant 3 are (-, -).
- Quadrant 4 coordinates are (+, -).
Step 6. Complete the goniometric circumference using the trick of the hand
Now you can fill in the coordinates of each quadrant, even when the angles are different. Remember to swap the positive and negative signs according to the quadrant.
Method 3 of 3: Use Fun Tricks to Memorize
Step 1. Sing a song about the goniometric circumference
You will more easily remember this information by putting a melody to it. You can choose a tune that you like and make up your own song, or learn one that someone else made up. Sing out loud to practice and then do it mentally when you need to remember the goniometric circumference.
Here's a song that might help you memorize (it's in English):
Step 2. Play an online game related to goniometric circumference
There are many online games that you can play for free. With them you can practice completing a goniometric circumference and at the same time have fun! In addition, you can study without getting bored. Here are some options:
- https://www.mathwarehouse.com/unit-circle/unit-circle-game.php
- https://www.sporcle.com/games/mhershfield/unit-circle-picture-click
- https://www.purposegames.com/game/unit-circle-quiz
Step 3. Use flashcards if you prefer to memorize facts
You can create your own flashcards or look them up online and study the information by quadrant or by angle measure. You may find it best to create a series of cards by separating the information in different ways.
Try Quizlet's pre-designed cards: https://quizlet.com/17071364/unit-circle-degreesradianssinecosine-flash-cards/ or https://quizlet.com/30187064/sin-cos-and-tan-of-the -unit-circle-radians-flash-cards /
Advice
- If you have a test or quiz on goniometric circumference, first draw a circle on scratch paper so you can refer to it when solving problems.
- It would be nice if you complete the goniometric circumference to practice. You can do it through an online game or print a template. Here's one:
- Memorizing the entire goniometric circumference takes time. Be patient.