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Trigonometry - Graphs of Trigonometric Functions

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    Study Guide Trigonometry – Graphs of Trigonometric Functions 1. Circular Functions 1. 1 The Unit Circle Figure 1 Unit circle reference. Figure 2 Range of values of trig functions.

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    Study Guide The equation draws a circle on the coordinate plane. This circle is called the unit circle . • Its center is at the origin ( 0,0 ) . • Its radius is 1 unit. The unit circle is extremely important because it helps us understand trigonometric and circular functions in a visual way. 1.2 From Angles to Numbers In regular trigonometry, functions like sine and cosine take angles as inputs. Their outputs are real numbers . Circular functions work a little differently: • Their inputs are real numbers (measured in radians ). • These numbers represent arc lengths along the unit circle. • Their outputs are still real numbers. They are called circular functions because the values come from moving around a circle . 1.3 How Radians Fit In Radians connect angles to circles naturally. • The radian measure of an angle is based on arc length , not degrees. • Because arc length depends on the circle, radians are perfect for circular functions. • On the unit circle , arc length and angle measure match exactly. This is why trigonometric functions defined on the unit circle lead directly to circular functions.

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    Study Guide 1.4 Defining Sine and Cosine Using the Unit Circle Start with the unit circle (x 2 + y 2 = 1). 1. Begin at point A(1, 0) on the positive x - axis. 2. Let (q) be any real number. 3. Move |q| units along the circle: o Counterclockwise if (q > 0) o Clockwise if (q < 0) 4. You arrive at a point (P(x, y ) . Now define: • sin q = y • cos q = x So, sine is the y - coordinate and cosine is the x - coordinate of point (P). 1.5 Other Circular (Trigonometric) Functions Once sine and cosine are defined, the other functions are built from them: These definitions match what you learned in trigonometry — just written in terms of circular functions. 1.6 Domain of Circular Functions Because you can move any distance around the unit circle: • The domain of sine and cosine is all real numbers . • Positive values move counterclockwise. • Negative values move clockwise. So, (sin q) and (cos q) exist for every real number (q) .

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    Study Guide 1.7 Range of Circular Functions While the domain is unlimited, the range is not. • Sine and cosine are coordinates on the unit circle. • All x - and y - values on the unit circle lie between −1 and 1. Therefore: This means the range of both sine and cosine is from −1 to 1. 1.8 Why Circular Functions Matter Circular functions: • Connect geometry and algebra • Explain why trigonometric values repeat • Make radians feel natural and meaningful • Are essential for advanced math, physics, and engineering Understanding them through the unit circle makes everything clearer and more intuitive. Example 1 : When Does the Sine Value Equal 1? Question: Which values of (x), between ( - 2 π) and (2 π), make the sine function equal to 1? In other words, we are looking for all values of (x) in this interval such that:

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    Study Guide Figure 3 Drawing for Example 1. Step 1: Think About the Unit Circle On the unit circle, the sine of an angle corresponds to the y - coordinate of a point on the circle. So asking when (sin x = 1) is the same as asking: When is the point on the unit circle at a height of 1 ? Step 2: Identify Where Sine Equals 1 The y - coordinate is equal to 1 only at the top of the unit circle , which is the point: This means we want all values of (x) that land us at this point. Step 3: Find the Correct Angle Measures Starting from ( 1, 0 ) and moving around the unit circle: • The first time we reach ( 0, 1 ) is at

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    Study Guide • If we continue moving around the circle clockwise, we reach the same point again at Both of these values lie between ( - 2 π) and (2 π ). Step 4: State the Final Answer The values of (x) for which the sine function has a value of 1 in the given interval are: Example 2 : When Does the Cosine Value Equal −1? Question: Which values of (x), between ( - 2 π) and (2 π), make the cosine function equal to ( - 1)? In mathematical terms, we are solving: Figure 4 Drawing for Example 2.

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    Study Guide Step 1: Connect Cosine to the Unit Circle On the unit circle, the cosine of an angle corresponds to the x - coordinate of a point on the circle. So asking when (cos x = - 1) is the same as asking: When is the point on the unit circle all the way to the left , at an x - value of ( - 1)? Step 2: Identify the Key Point The x - coordinate equals ( - 1) at exactly one point on the unit circle: This is the leftmost point on the circle. Step 3: Find the Angle Values Starting from ( 1, 0 ) and moving around the unit circle: • Moving counterclockwise, you reach ( - 1, 0 ) at • Moving clockwise, you reach the same point at Both values are within the interval ( - 2 π ≤ x ≤ 2 π ). Step 4: State the Final Answer The values of (x) for which the cosine function has a value of ( - 1) in the given interval are:

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    Study Guide Example 3 : Finding All Six Circular Functions Given: The point lies on the unit circle . The arc length from point (A(1,0 ) to point (P) is (q) units. Question: What are the values of the six circular (trigonometric) functions of (q)? Step 1: Use the Coordinates of the Point On the unit circle: • The x - coordinate gives the cosine value • The y - coordinate gives the sine value So, directly from point (P): Step 2: Find Tangent Tangent is defined as sine divided by cosine: Substitute the values:

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    Study Guide Step 3: Find Cotangent Cotangent is the reciprocal of tangent (or cosine divided by sine): Step 4: Find Secant Secant is the reciprocal of cosine: Step 5: Find Cosecant Cosecant is the reciprocal of sine: Step 6: Summary of All Six Values

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    Study Guide Step 7: Understanding the Signs The signs of the trigonometric functions depend on which quadrant the point lies in. Summary • Sine represents the vertical position on the unit circle. • The maximum value of sine is 1 . • This value occurs whenever the angle places the point at the top of the circle . • Cosine represents the horizontal position on the unit circle. • The minimum value of cosine is − 1 .

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