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Perplex
Perplex
Dashboard
Topics
Exponents & LogarithmsRounding & ErrorSequences & SeriesCounting & BinomialsProof and ReasoningComplex NumbersAlgebra Skills
Cartesian plane & linesQuadraticsFunction TheoryTransformations & asymptotesPolynomials
2D & 3D GeometryTrig equations & identitiesVectors
ProbabilityDescriptive StatisticsBivariate StatisticsDistributions & Random Variables
DifferentiationIntegrationDifferential EquationsMaclaurin
Review VideosFormula BookletMy Progress
BlogLanding Page
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Perplex

Trig equations & identities (Lesson 2/8)

The Unit Circle

1 / 25

Discussion

Imagine we have a circle centered about the origin ​O(0,0)​ with radius ​1, and we place a point ​P​ on its circumference such that the line ​(OP)​ forms an acute angle of ​30°​ with the ​x​-axis:


(a)

Find the coordinates of ​P.

Solution:

The hypotenuse of the triangle is the radius of the circle, which is 1. The triangle is placed so that its hypotenuse makes a ​30∘​ angle with the ​x​-axis, starting from the origin.


To find the coordinates where the triangle (hypotenuse) meets the circle, we use the definitions of sine and cosine for a right triangle:

​
sinθ=hypotenuseopposite​cosθ=hypotenuseadjacent​
​

These are the SOH and CAH formulas.

Since the hypotenuse is 1, the coordinates of the point on the circle at a ​30∘​ angle from the ​x​-axis are:

​
(cos(30∘),sin(30∘))
​


Evaluate ​cos(30∘)​ and ​sin(30∘)​ using a calculator:

​
cos(30∘)sin(30°)​≈0.866(which is actually 2√3​)=0.5​
​

So, the intersection point is:

​
(0.866,0.5)
​
1 free

Trig equations & identities (Lesson 2/8)

The Unit Circle

1 / 25

Discussion

Imagine we have a circle centered about the origin ​O(0,0)​ with radius ​1, and we place a point ​P​ on its circumference such that the line ​(OP)​ forms an acute angle of ​30°​ with the ​x​-axis:


(a)

Find the coordinates of ​P.

Solution:

The hypotenuse of the triangle is the radius of the circle, which is 1. The triangle is placed so that its hypotenuse makes a ​30∘​ angle with the ​x​-axis, starting from the origin.


To find the coordinates where the triangle (hypotenuse) meets the circle, we use the definitions of sine and cosine for a right triangle:

​
sinθ=hypotenuseopposite​cosθ=hypotenuseadjacent​
​

These are the SOH and CAH formulas.

Since the hypotenuse is 1, the coordinates of the point on the circle at a ​30∘​ angle from the ​x​-axis are:

​
(cos(30∘),sin(30∘))
​


Evaluate ​cos(30∘)​ and ​sin(30∘)​ using a calculator:

​
cos(30∘)sin(30°)​≈0.866(which is actually 2√3​)=0.5​
​

So, the intersection point is:

​
(0.866,0.5)
​
1 free