Values on the Unit Circle
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Question 1 of 7
1. Question
Find the value of sin60°- 1.
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Coordinates of Trigonometric Functions
(cosθ,sinθ)=(x,y)There are known values for a specific set of trigonometric functions within all quadrants that follows a pattern.Trigonometric Functions
sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacentMethod OneThe coordinates of each specific value will be a fraction with a denominator of 2The x-coordinate’s numerator will be the equivalent of the square root of 1,2, and 3 respectively for each quadrant, starting from the value nearest to the y-axis to the value nearest to the x-axisThe y-coordinate’s numerator will be the equivalent of the square root of 1,2, and 3 respectively for each quadrant, starting from the value nearest to the x-axis to the value nearest to the y-axisRemember to apply the proper signs for each value depending on the quadrant they are in1st Quadrant(Q1) = (+,+) 2nd Quadrant(Q2) = (-,+) 3rd Quadrant(Q3) = (-,-) 4th Quadrant(Q4) = (+,-) Given that (cosθ,sinθ)=(x,y), sin60° will be the y-coordinate of 60°, which is √32sin60°=√32Method TwoWe can use special triangles to solve this problem.Since the given angle measures 60°, we can use the 30-60-90 triangle.To solve for sin60°, we can use the known values of the side opposite to it and the hypotenuse.Since we have the opposite and hypotenuse values, we can solve for sin60°sin60° = oppositehypotenuse sin60° = √32 sin60°=√32 -
Question 2 of 7
2. Question
Find the radian value of 330°-
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There are known values for a specific set of trigonometric functions within all quadrants that follows a pattern.Method OneKeep in mind that 180°=πGiven that value, we can easily compute for the radian values of 30°,45°, and 60°30°×π180° = 30°π180° = π6 45°×π180° = 45°π180° = π4 60°×π180° = 60°π180° = π3 For the second quadrant, you can get the radian value of 120°,135°, and 150° by using the same radian value of their parallels on the first quadrant and changing the numerator’s constant to the difference of their numerator and denominator120°∣∣60°120° = (3-1)π3 = 2π3 135°∣∣45°135° = (4-1)π4 = 3π4 150°∣∣30°150° = (6-1)π6 = 5π6 For the third and fourth quadrant, you can get the radian value of 210°,225°,240°,300°,315° and 330° by using the same radian value of their opposites on the first and second quadrant and adding the value of their denominator to their numerator’s constant210°↔30°210° = (1+6)π6 = 7π6 225°↔45°225° = (1+4)π4 = 5π4 240°↔60°240° = (1+3)π3 = 4π3 300°↔120°300° = (2+3)π3 = 5π3 315°↔135°315° = (3+4)π4 = 7π4 330°↔150°330° = (5+6)π6 = 11π6 Given these known values, the radian value of 330° would be 11π6330°=11π6Method TwoSince we know that 180°=π, we can easily get the radian value of 1°180° = π 180°180° = π180° Divide both sides by 180° 1° = π180° Now that we know the radian value of 1°, we can multiply 330° to it in order to get its radian value.1° = π180° 1°×330° = π180°×330° Multiply both sides by 330° 330° = 330°π180° 330° = 330°π÷30°180÷30° Divide the numerator and denominator by 30° 330° = 11π6 330°=11π6 -
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Question 3 of 7
3. Question
Given the image below, find the value of:(i) sin90°(ii) cos180°(iii) tan360°-
(i) sin90°= (1)(ii) cos180°= (-1)(iii) tan360°= (0)
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Trigonometric Functions
sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacentUsing the given image, we can see that the radius of the circle will be 1. Therefore, the circle follows the formula x2+y2=1, where 1 is the radius.Make a reference triangle by making a line from the point of origin (0,0) going to any point in the circle to represent the radius and connect it to either the x or y-axisNotice that, given angle θ, the circle has the vertical line parallel to the y-axis as its opposite side and the line on the x-axis as the adjacent sideNow that we have the opposite and adjacent sides and the radius 1 as the hypotenuse, we can use the trigonometric functions to find their respective valuessin = oppositehypotenuse = y1 = y cos = adjacenthypotenuse = x1 = x tan = oppositeadjacent = yx x and y are both coordinate valuesGiven the following values, we can use the image as reference and solve for sin90°,cos180°, and tan360°90°=(0,1)sin90° = y = 1 180°=(-1,0)cos180° = x = -1 360°=(1,0)tan360° = yx = 01 = 0 (i)sin90°=1(ii)cos180°=-1(iii)tan360°=0 -
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Question 4 of 7
4. Question
Find the value of tan(7π6)-
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Converting Radian to Degrees
degrees=radian×180°πCoordinates of Trigonometric Functions
(cosθ,sinθ)=(x,y)Trigonometric Functions
sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacentThere are known values for a specific set of trigonometric functions within all quadrants that follows a pattern.Method OneFirst, convert the radian to degreesdegrees = radians×180°π = 7π6×180°π = 1260°6 ππ=1 = 210° Next, recall the values of the trigonometric functionsGiven that (cosθ,sinθ)=(x,y), we are given the following valuessin210° = -12 cos210° = -√32 Finally, solve for the value of tan210°tan210° = sin210°cos210° = -12-√32 Substitute known values = 1√3 Simplify tan(7π6) = 1√3 tan(7π6)=1√3Method TwoFirst, convert the radian to degreesdegrees = radians×180°π = 7π6×180°π = 1260°6 ππ=1 = 210° Make a reference triangle by making a line from the point of origin (0,0) going to any point in the circle that would represent the angle 210° and connect it to either the x or y-axisFrom here, we can see that 210° lies on the 3rd quadrant, which means tan210° is positive.To find θ, we can subtract 180° (the horizontal line) from 210°.210°-180°=30°This means we can also refer to tan(7π6) as tan30°Knowing that the reference triangle has 30° and 90° angles, we can use the special 30-60-90 triangle.To solve for tan30°, we can use the known values of the sides opposite and adjacent to it.Since we have the opposite and adjacent values, we can solve for tan30°tan30° = oppositeadjacent tan30° = 1√3 tan(7π6) = 1√3 tan(7π6)=1√3 -
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Question 5 of 7
5. Question
Find the value of tan(5π3)-
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Converting Radian to Degrees
degrees=radian×180°πCoordinates of Trigonometric Functions
(cosθ,sinθ)=(x,y)Trigonometric Functions
sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacentThere are known values for a specific set of trigonometric functions within all quadrants that follows a pattern.Method OneFirst, convert the radian to degreesdegrees = radians×180°π = 5π3×180°π = 900°3 ππ=1 = 300° Alternatively, we can convert the value to a mixed fraction.5π3 = 123π Knowing these values, we can identify the quadrant that contains 300° or 5π3 using this chart:Note that if you prefer using the mixed number form of the radian, you will also try getting the estimated location by finding the quadrant beyond π and 23 to 2πThe value should be located in the fourth quadrant.Now use a special triangle to find the value of 5π3Note that 5π3 can be written as 5×π3 so we can use the π3 value as a reference angle.tan300° = oppositeadjacent = √31 Substitute known values = √3 Simplify Finally, keep the following in mind when providing the proper value of trigonometric functions for each quadrant1st Quadrant = All are positive 2nd Quadrant = Only sin is positive 3rd Quadrant = Only tan is positive 4th Quadrant = Only cos is positive Therefore, the known value of tan5π3=-√3tan(5π3)=-√3Method TwoFirst, convert the radian to degreesdegrees = radians×180°π = 5π3×180°π = 900°3 ππ=1 = 300° Make a reference triangle by making a line from the point of origin (0,0) going to any point in the circle that would represent the angle 300° and connect it to either the x or y-axisFrom here, we can see that 300° lies on the 4th quadrant, which means tan300° is negative.To find θ, we can subtract 300° from 360° (the horizontal line).360°-300°=60°This means we can also refer to tan(5π3) as tan60°Knowing that the reference triangle has 60° and 90° angles, we can use the special 30-60-90 triangle.To solve for tan60°, we can use the known values of the sides opposite and adjacent to it.Since we have the opposite and adjacent values, we can solve for tan60°tan60° = oppositeadjacent tan60° = √31 tan(5π3) = √3 Recall that the angle lies on the 4th quadrant, so the answer should be negative.tan(5π3) = -√3 tan(5π3)=-√3 -
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Question 6 of 7
6. Question
Find the value of sin(5π4)-
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Coordinates of Trigonometric Functions
(cosθ,sinθ)=(x,y)There are known values for a specific set of trigonometric functions within all quadrants that follows a pattern.Trigonometric Functions
sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacentValues of a Right Triangle in a Unit Circle
Method OneFirst, find the acute reference angle by finding the value to be added to π to get 5π4.π can be written as 4π4. So if we subtract this value from 5π4, we can get the acute reference angle.5π4-4π4 = π4 Therefore, 5π4 can be written as π+π4 with π4 as the reference angle.Next, we can identify the quadrant that contains 5π4 using this chart:The value should be located in the third quadrant.Now use a special triangle that has the reference angle to find the value of 5π4sinπ4 = oppositehypotenuse = 1√2 Substitute known values Finally, keep the following in mind when providing the proper value of trigonometric functions for each quadrant1st Quadrant = All are positive 2nd Quadrant = Only sin is positive 3rd Quadrant = Only tan is positive 4th Quadrant = Only cos is positive Therefore, the known value of sin5π4=-1√2sin5π4=-1√2Method TwoFirst, convert the radian to degreesdegrees = radians×180°π = 5π4×180°π = 900°4 ππ=1 = 225° Make a reference triangle by making a line from the point of origin (0,0) going to any point in the circle that would represent the angle 225° and connect it to either the x or y-axisFrom here, we can see that 225° lies on the 3rd quadrant, which means sin225° is negative.To find θ, we can subtract 180° (the horizontal line) from 225°.225°-180°=45°This means we can also refer to sin(5π4) as sin45°Knowing that the reference triangle has 45° and 90° angles, we can use the special 45-45-90 triangle.To solve for sin45°, we can use the known values of the side opposite to it and the hypotenuse.Since we have the opposite and hypotenuse values, we can solve for sin45°sin45° = oppositehypotenuse sin45° = 1√2 sin(5π4) = 1√2 Recall that the angle lies on the 3rd quadrant, so the answer should be negative.sin(5π4) = −1√2 sin5π4=-1√2 -
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Question 7 of 7
7. Question
Find the value of cos(2π3)-
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Chapters- Chapters
Coordinates of Trigonometric Functions
(cosθ,sinθ)=(x,y)There are known values for a specific set of trigonometric functions within all quadrants that follows a pattern.Trigonometric Functions
sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacentValues of a Right Triangle in a Unit Circle
Method OneFirst, you can convert the radian to degreesdegrees = radians×180°π = 2π3×180°π = 360°3 ππ=1 = 120° Knowing these values, we can identify the quadrant that contains 120° or 2π3 using this chart:The value should be located in the second quadrant.Next, find the acute reference angle by finding the value to be subtracted from π to get 2π3.π can be written as 3π3. So if we subtract 2π3 from this value, we can get the acute reference angle.3π3-2π3 = π3 Therefore, 2π3 can be written as π-π3 with π3 as the reference angle..Now use a special triangle that has the reference angle to find the value of 2π3cosπ3 = adjacenthypotenuse = 12 Substitute known values Finally, keep the following in mind when providing the proper value of trigonometric functions for each quadrant1st Quadrant = All are positive 2nd Quadrant = Only sin is positive 3rd Quadrant = Only tan is positive 4th Quadrant = Only cos is positive Therefore, the known value of cos2π3=-12cos2π3=-12Method TwoFirst, convert the radian to degreesdegrees = radians×180°π = 2π3×180°π = 360°3 ππ=1 = 120° Make a reference triangle by making a line from the point of origin (0,0) going to any point in the circle that would represent the angle 120° and connect it to either the x or y-axisFrom here, we can see that 120° lies on the 2nd quadrant, which means cos120° is negative.To find θ, we can subtract 120° (the horizontal line) from 180°.180°-120°=60°This means we can also refer to cos(2π3) as cos60°Knowing that the reference triangle has 60° and 90° angles, we can use the special 30-60-90 triangle.To solve for cos60°, we can use the known values of the side adjacent to it and the hypotenuse.Since we have the adjacent and hypotenuse values, we can solve for cos60°cos60° = adjacenthypotenuse cos60° = 12 cos(2π3) = 12 Recall that the angle lies on the 2nd quadrant, so the answer should be negative.cos(2π3) = −12 cos2π3=-12 -
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Quizzes
- Converting Angle Measures 1
- Converting Angle Measures 2
- Converting Angle Measures 3
- Finding the Central Angle in a Circle
- Finding Areas in a Circle
- Values on the Unit Circle
- Finding Missing Angles Using the Unit Circle
- Trigonometric Ratios in the Unit Circle
- Trig Exact Values 1
- Trig Exact Values 2
- Trig Equations
- Derivative of a Trigonometric Function 1
- Derivative of a Trigonometric Function 2
- Derivative of a Trigonometric Function 3
- Applications of Differentiation
- Integral of a Trigonometric Function 1
- Integral of a Trigonometric Function 2
- Applications of Integration
- Graphing Trigonometric Functions 1
- Graphing Trigonometric Functions 2
- Graphing Trigonometric Functions 3
- Graphing Trigonometric Functions 4