Unit 4 - Circular Functions
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Lesson 4.1 - Radian Measure
Learning Objectives: SWBAT
Learning Objectives: SWBAT
- Explain the difference between degree and radian measure of angles
- Draw rough sketch of reference angles given in degrees or radians
- Convert angle measures from degrees to radians and vice versa
- Radians are simply another way of measuring angles (instead of degrees)
- Radian measure is based on the relationship between the angle and the arc it creates
- We know the angle that creates one full revolution of a circle is 360 degrees
- We also know that the circumference (entire arc) of a circle when the radius of the circle = 1 is 2 pi
- Therefore, 360 degrees = 2 pi radians.
- From this, we can determine a quick formula to convert degrees to radians and vice versa
- To go from degrees to radians, multiply the degrees by pi/180
- To go from radians to degrees multiply the radians by 180/pi
- Remember, when we make an angle from standard position, we create the angle in a counterclockwise direction
- In this lesson we also introduce the concept of NEGATIVE angles in which the angle is created in a clockwise directions
- So, a positive angle of 90 degrees ends up in the same "location" as a negative angle of 270 degrees
- The diagram on the bottom of page 2 of the lesson provides you with the most common angles measured in radians. Eventually, you will need know this by heart
- CLICK HERE for a video that defines radian angle measures and demonstrates the relationship radian's have to the arc length created by a central angle
- CLICK HERE for a video that demonstrates how to sketch angles in radians (including negative angles)
- CLICK HERE for a video that demonstrates how to convert from degrees to radians and vice versa.
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Lesson 4.2 - Coterminal Angles
Learning Objectives: SWBAT
Learning Objectives: SWBAT
- Determine and sketch angles that are coterminal to a given angle
- Determine complementary and supplementary angles (in radians)
- This lesson is fairly short and has three key points:
- Coterminal angles are angles that have the same initial and terminal side. For example, 3∏/2 and -∏/2 are coterminal. We also discussed how there are an infinite number of coterminal angles for a given angle since we can always add/subtract another revolution (2∏ around the circle).
- Complementary angles are angles that add to right angles. Previously a right angle was alway measured as 90 degrees, however now, we are measuring right angles as ∏/2 radians. Remember, angles that are > than right angles do not have compliments
- Supplementary angles are angles that add to 180 degrees or ∏ radians. Remember, angles that are > than 180 degrees (or ∏ radians) do not have supplements.
- CLICK HERE for a video that demonstrates how to find coterminal angles of a given angle
- CLICK HERE for a video that demonstrates how to determine complementary/supplementary angles given in radians
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Lesson 4.3 - Evaluating Trig Functions (part 1)
Learning Objectives: SWBAT
Learning Objectives: SWBAT
- Explain the relationship between the "reciprocal" Trig functions (Secant, Cosecant, Cotangent) and "non-reciprocal" functions (Cosine, Sine and Sine)
- Evaluate the six trigonometric functions given any point/angle
- Evaluate the other 5 trig functions given the value of one trig function and its quadrant.
- The first page of the file has a "prelude" to the actual lesson in which we introduce the idea of "reciprocal Trig functions"
- If you look at the diagram on this page, you will see the SOH CAH TOA equations for all right triangles.
- The reciprocal functions simply "flip" the combination of "opposite/adjacent/Hypotenuse" upside down (this is why they are called reciprocal functions). Here is what you need to remember
- Secant is the reciprocal of cosine
- Cosecant is the reciprocal of sine
- Cotangent is the reciprocal of tangent
- Therefore, if you know Cosine, sine and tangent, you also know Secant, Cosecant and Cotangent (just flip upside down)
- The bottom of the prelude and the blue box on page one of the lesson are two different ways of showing the same information. As mentioned above, the prelude relates the ratios to SOH CAH TOA., the blue box relates the ratios to (x, y) coordinates created by an angle
- The x coordinate is the adjacent side of a right triangle that is drawn from the terminal point to the x axis, therefore it relates to COSINE
- The y coordinate is the opposite side of a right triangle that is drawn from the terminal point to the x axis, therefore it relates to SINE
- "r" is the radius of the circle that is created by the (x, y) coordinate and always starts at (0, 0). This is the hypotenuse of a right triangle that is drawn from the terminal point to the x axis. "r" is equivalent to the magnitude of the vector that is created by the terminal point.
- It is important to notice that in each case, the right triangle is ALWAYS drawn to the x axis. It will NEVER be drawn to the y axis, regardless of the quadrant that the point is in (lesson 4.4 will further reinforce this when you review reference angles).
- The example of the bottom of the first page of the lesson is a good and basic illustration of how this works. The example only gives the three "non-reciprocal" functions but simply flip these upside down to get the three "reciprocal" functions.
- The second video below has several other examples, please make sure you watch the entire video especially the last example where the triangle has square roots as part of the sides. Notice how square roots/fractions need to be simplified and denominators need to be rationalized. This will happen very often, you need to be able to do this part.
- CLICK HERE for a video that provides a conceptual overview of the reciprocal functions.
- CLICK HERE for a video that shows how to evaluate the six trig functions given a point
- The key to this lesson is the blue box on the first page where you have all of the combinations of "x", "y" and "r"
- The quadrant that is given in these problems tells you the signs of x and y (r is always positive).
- The trig function given will give you two of the three "x", "y" and "r" values. It is up to you to determine the third value via the paythagorean theorem
- Remember, x and y are the legs, and r is the hypotenuse
- Once you have identified "x", "y" and "r" (remember, you are given two and will determine the third), all you doing here is re-arranging these values as laid out in the blue box on the front page
- The signs of these values is determined by the quadrant that is given. Here are some tips
- Cosecant and Secant are always the same sign as sine and cosine respectively
- If sine and cosine are both the same sign, then tangent and cotangent will always be positive
- If sine and cosine are different signs, then tangent and cotangent will always be negative
- CLICK HERE for a video that demonstrates how to find the five other trig functions given one trig function and its quadrant (this example gives a cosine value)
- CLICK HERE for a video that demonstrates how to find the five other trig functions given one trig function and its quadrant (this example gives a tangent value)
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Lesson 4.4 - Reference Angles
Learning Objectives: SWBAT
Learning Objectives: SWBAT
- Determine and sketch reference angles of a given angles
- This lesson is fairly short. In lesson 3.14, we learned to sketch direction angles. Reference angles are different from direction angles in the following ways:
- Reference angles are ALWAYS acute
- Reference angles are ALWAYS positive
- Reference angles are ALWAYS bound by the x axis and another terminal side that is in one of the four quadrants. (NEVER the y axis). This means that the right triangle that is created by the reference angle will always have the right angle ON the x axis
- There are several ways to determine the measure of a reference angle but the most common way is going to usually involve subtracting from one of the quadrant boundary angles (90, 180, 270, 360 degrees or pi/2, pi, 3pi/2, 2 pi radians)
- As an example, an angle of 210 degrees will have a reference angle of 30 degrees.
- Using the same example above (but in radians) an angle of 7pi/6 radians will have a reference angle of pi/6 radians
- The example below the blue box on page one of the lesson has several good visuals of these angles
- CLICK HERE for a video that models examples of how to find reference angles of a given angle (examples in both degrees and radians)
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Evaluating Unit Circle Angles Activity |
Activity - Evaluating the Unit Circle
Learning Objective: SWBAT
Learning Objective: SWBAT
- Determine the sine, cosine, tangent, cosecant, secant and cotangent for all common angles of the unit circle
- The common angles of the unit circle are based on "special right triangles".
- In Quadrant 1, the common angles are 30 degrees (pi/6), 45 degrees (pi/4), 60 degrees (pi/3) and 90 degrees (pi/2)
- In this activity, you will use the properties of special right triangles to determine the sine, cosine, tangent, cosecant, secant and cotangent (the six trig functions) of each angle. Here is what you need to know:
- To start, remember, that the radius of the unit circle is always 1. This is key. This is the hypotenuse of the right triangle created by the reference angle.
- Always start the process of finding all six functions by finding SINE and COSINE. All of the other functions are "born" from these two.
- The x coordinate of each angle is COSINE (because it is the adjacent side of the right triangle created by the reference angle)
- The y coordinate of each angle is SINE (because it is the opposite side of the right triangle created by the reference angle
- As an example, for a 30 degree (pi/6 radians) angle in quadrant 1. Using special right triangle rules, we divide the hypotenuse by 2 to determine the short (in this case opposite) side. Therefore the SINE of 30 degrees (pi/6 radians) = 1/2
- Then, also using special right triangle rules, we multiply the "short" side by sqrt 3 to determine the "long" side. If we multiply the short side (1/2) by sqrt of 3, the result is (sqrt 3)/2. Therefore the COSINE of 30 degrees (pi/6 radians) = (sqrt 3)/2.
- To determine TANGENT, we know that it is opposite/adjacent. Since opposite = SINE and adjacent = COSINE another way of looking at TANGENT is that it = SINE/COSINE. So, we need to use the sine/cosine values determined in our previous steps to determine tangent. We will divide sine (1/2) by cosine (sqrt 3)/2. REMEMBER, when dividing fractions you multiply by the reciprocal. After you do this you do this you will be left with 1/(sqrt 3). After rationalizing the denominator the final result for the TANGENT of 30 degrees (pi/6 radians) = (sqrt 3)/3
- To determine COSECANT, SECANT and COTANGENT, simply take the reciprocals of SINE, COSINE and Tangent respectively. Remember, you MUST rationalize any denominators that have radical numbers in them.
- As you progress through the activity, you will realize that the corresponding angles in quadrant 2 have the same numerical values as the angles in quadrant 1, however the (+/-) signs will be different depending on the quadrant. You will also notice that there is symmetry to these values throughout the circle.
- I have completed the first column of the chart for you based on the information in the example above, your job is to determine/fill in the rest of the chart
- CLICK HERE for a video that demonstrates how to evaluate sine, cosine and tangent for a 30 degree (pi/6 radians) angle on the unit circle similar to the example above
- CLICK HERE for a video that demonstrates how to evaluate sine, cosine and tangent for all common angles of the unit circle
- CLICK HERE for a video that demonstrates how to evaluate all six trigonometric functions for all common angles of the unit circle
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Lesson 4.5 - Evaluating Trig Functions (part 2)
Learning Objectives: SWBAT
Learning Objectives: SWBAT
- Evaluate trigonometric functions for common angles on the unit circle
- This lesson is an extension from the previous activity
- You will be given a variety of angles both in degrees and radians.
- Sketch the angle and determine the quadrant that the angle is in. This will determine the signs of the each trig function value
- Use the process as in the activity to evaluate sine, cosine and tangent. The problems do not ask you to evaluate cosecant, secant and cotangent but you should still be able to do it as in lesson 4.5 (part 1)
- CLICK HERE for a video that demonstrates how to evaluate trig functions for angles on the unit circle