Trigonometric Functions: Unit Circle Approach - Revision history

Lila: /* Resources */

2021-10-25T18:53:51Z

Resources

Lila at 20:57, 5 October 2021

2021-10-05T20:57:09Z

Lila at 20:53, 5 October 2021

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Lila at 20:51, 5 October 2021

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Lila at 20:48, 5 October 2021

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Lila at 20:41, 5 October 2021

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Lila: /* Example 2 */

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Example 2

← Older revision		Revision as of 18:53, 25 October 2021
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	* [https://en.wikibooks.org/wiki/High_School_Trigonometry/Defining_Trigonometric_Functions Defining Trigonometric Functions], WikiBooks: High School Trigonometry		* [https://en.wikibooks.org/wiki/High_School_Trigonometry/Defining_Trigonometric_Functions Defining Trigonometric Functions], WikiBooks: High School Trigonometry
	* [https://en.wikibooks.org/wiki/Trigonometry/The_Unit_Circle The Unit Circle], WikiBooks: Trigonometry		* [https://en.wikibooks.org/wiki/Trigonometry/The_Unit_Circle The Unit Circle], WikiBooks: Trigonometry
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Trigonometry/The_Unit_Circle The Unit Circle, WikiBooks: Trigonometry] under a CC BY-SA license
		+	* [https://en.wikibooks.org/wiki/High_School_Trigonometry/Defining_Trigonometric_Functions Defining Trigonometric Functions, WikiBooks: High School Trigonometry] under a CC BY-SA license

← Older revision		Revision as of 20:57, 5 October 2021
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	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Trigonometric%20Functions_%20Unit%20Circle%20Approach/Esparza%201093%20Notes%202.2.pdf Trigonometric Functions: Unit Circle Approach]. Written notes created by Professor Esparza, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Trigonometric%20Functions_%20Unit%20Circle%20Approach/Esparza%201093%20Notes%202.2.pdf Trigonometric Functions: Unit Circle Approach]. Written notes created by Professor Esparza, UTSA.
	* [https://en.wikibooks.org/wiki/High_School_Trigonometry/Defining_Trigonometric_Functions Defining Trigonometric Functions], WikiBooks: High School Trigonometry		* [https://en.wikibooks.org/wiki/High_School_Trigonometry/Defining_Trigonometric_Functions Defining Trigonometric Functions], WikiBooks: High School Trigonometry
		+	* [https://en.wikibooks.org/wiki/Trigonometry/The_Unit_Circle The Unit Circle], WikiBooks: Trigonometry

← Older revision		Revision as of 20:53, 5 October 2021
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−	We can use the figure above to determine values of the trig functions for the quadrantal angles. For example, <math>\sin~~(90°)</math>~~ = y = 1</math>. <math>\sec~~(45°)~~ = \frac{1}{x} = \frac{2}{\sqrt{2}} = \sqrt{2}</math>	+	We can use the figure above to determine values of the trig functions for the quadrantal angles. For example, <math>\sin{90^\circ} = y = 1</math>. <math>\sec{45^\circ} = \frac{1}{x} = \frac{2}{\sqrt{2}} = \sqrt{2}</math>.

	===Example 1===		===Example 1===

← Older revision		Revision as of 20:51, 5 October 2021
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−	We can use the figure above to determine values of the trig functions for the quadrantal angles~~. The x-coordinate at each point on the circle represents the value of cosine for the specific angle, and the y-coordinate is sine~~. For example, sin(90°) = ''y'' = 1.	+	We can use the figure above to determine values of the trig functions for the quadrantal angles. For example, <math>\sin(90°)</math> = y = 1</math>. <math>\sec(45°) = \frac{1}{x} = \frac{2}{\sqrt{2}} = \sqrt{2}</math>
−
−	~~We can figure out the other trigonometric functions by rewriting them in terms of sine and cosine.~~
−
−	* <math> \~~tan\theta~~ = \frac{y}{x} = \frac{y}{x}\~~frac~~{r}{r} = \~~frac~~{~~y}{r}\div\frac{x}{r} = \frac{\sin\theta}{\cos\theta~~}</math>
−	*
−	*
−	*

	===Example 1===		===Example 1===

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−	We can use the figure above to determine values of the trig functions for the quadrantal angles. For example, sin(90°) = ''y'' = 1.	+	We can use the figure above to determine values of the trig functions for the quadrantal angles. The x-coordinate at each point on the circle represents the value of cosine for the specific angle, and the y-coordinate is sine. For example, sin(90°) = ''y'' = 1.
		+
		+	We can figure out the other trigonometric functions by rewriting them in terms of sine and cosine.
		+
		+	* <math> \tan\theta = \frac{y}{x} = \frac{y}{x}\frac{r}{r} = \frac{y}{r}\div\frac{x}{r} = \frac{\sin\theta}{\cos\theta}</math>
		+	*
		+	*
		+	*

	===Example 1===		===Example 1===

@@ Line 15: / Line 15: @@
 Notice that in the unit circle, the sine and cosine of an angle are the ''x'' and ''y'' coordinates of the point on the terminal side of the angle. Now we can find the values of the trigonometric functions of any angle of rotation, even the quadrantal angles, which are not angles in triangles.
 [[File:Unit Circle Definitions of Six Trigonometric Functions.png|The center of the unit circle is the origin <math>O</math>, point <math>A=(x_A,y_A)</math>. Creating line segments starting from point <math>A</math> to <math>x</math> on the <math>x</math>-axis will create two line segments. From this, we learn that, in terms of <math>\theta</math>, point <math>A=(\cos\theta,\sin\theta)</math>and other properties.]]
 The center of the unit circle is the origin <math>O</math>, point <math>A=(x_A,y_A)</math>. Creating line segments starting from point <math>A</math> to <math>x</math> on the <math>x</math>-axis will create two line segments. From this, we learn that, in terms of <math>\theta</math>, point <math>A=(\cos\theta,\sin\theta)</math>and other properties.
-[[File:Unit_circle_angles.svg|thumb|397x397px|left|Labeled unit circle]]
 We can use the figure above to determine values of the trig functions for the quadrantal angles. For example, sin(90°) = ''y'' = 1.

@@ Line 18: / Line 18: @@
 [[File:Unit Circle Definitions of Six Trigonometric Functions.png|The center of the unit circle is the origin <math>O</math>, point <math>A=(x_A,y_A)</math>. Creating line segments starting from point <math>A</math> to <math>x</math> on the <math>x</math>-axis will create two line segments. From this, we learn that, in terms of <math>\theta</math>, point <math>A=(\cos\theta,\sin\theta)</math>and other properties.]]
-:[[File:Defining Trigonometric Functions Figure 5.svg|500px]]
+The center of the unit circle is the origin <math>O</math>, point <math>A=(x_A,y_A)</math>. Creating line segments starting from point <math>A</math> to <math>x</math> on the <math>x</math>-axis will create two line segments. From this, we learn that, in terms of <math>\theta</math>, point <math>A=(\cos\theta,\sin\theta)</math>and other properties.
 We can use the figure above to determine values of the trig functions for the quadrantal angles. For example, sin(90°) = ''y'' = 1.
@@ Line 45: / Line 47: @@
 There are several important angles in the unit circle that you will work with extensively in your study of trigonometry: 30°, 45°, and 60°. To find the values of the trigonometric functions of these angles, we need to know the ordered pairs. Let's begin with 30°.
 The terminal side of the angle intersects the unit circle at the point <math>\tfrac{\sqrt{3}}{2}, \tfrac{1}{2}</math>. (You will prove this in one of

@@ Line 15: / Line 15: @@
 Notice that in the unit circle, the sine and cosine of an angle are the ''x'' and ''y'' coordinates of the point on the terminal side of the angle. Now we can find the values of the trigonometric functions of any angle of rotation, even the quadrantal angles, which are not angles in triangles.
 :[[File:Defining Trigonometric Functions Figure 5.svg|500px]]
@@ Line 90: / Line 92: @@
 : The tangent value is the ratio of the ''y'' coordinate to the ''x'' coordinate. Because the ''x'' and ''y'' coordinates are the same for this angle, the tangent ratio is 1.
-[[File:Unit Circle Definitions of Six Trigonometric Functions.png|thumb|264x264px|The center of the unit circle is the origin <math>O</math>, point <math>A=(x_A,y_A)</math>. Creating line segments starting from point <math>A</math> to <math>x</math> on the <math>x</math>-axis will create two line segments. From this, we learn that, in terms of <math>\theta</math>, point <math>A=(\cos\theta,\sin\theta)</math>and other properties.]]
 ==Resources==
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Trigonometric%20Functions_%20Unit%20Circle%20Approach/Esparza%201093%20Notes%202.2.pdf Trigonometric Functions: Unit Circle Approach]. Written notes created by Professor Esparza, UTSA.
 * [https://en.wikibooks.org/wiki/High_School_Trigonometry/Defining_Trigonometric_Functions Defining Trigonometric Functions], WikiBooks: High School Trigonometry

@@ Line 41: / Line 41: @@
 c. sec 0° = 1
 : The ordered pair for this angle is (1, 0).  The ratio is <math>\tfrac{1}{x}</math> is <math>\tfrac{1}{1}</math> = 1.
 There are several important angles in the unit circle that you will work with extensively in your study of trigonometry: 30°, 45°, and 60°. To find the values of the trigonometric functions of these angles, we need to know the ordered pairs. Let's begin with 30°.
@@ Line 90: / Line 89: @@
 c. tan (45°) = 1
 : The tangent value is the ratio of the ''y'' coordinate to the ''x'' coordinate. Because the ''x'' and ''y'' coordinates are the same for this angle, the tangent ratio is 1.
 ==Resources==
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Trigonometric%20Functions_%20Unit%20Circle%20Approach/Esparza%201093%20Notes%202.2.pdf Trigonometric Functions: Unit Circle Approach]. Written notes created by Professor Esparza, UTSA.