Why (cos θ, sin θ)
Part of the Angles and the Unit Circle section of Coddy's Trigonometry journey. Lesson 25 of 57.
The circle of radius 1 centred at the origin is the unit circle. Drop a line from the point at θ to the x-axis and you get a right triangle whose hypotenuse is the radius, 1.
Hypotenuse 1: the legs are cos 50° and sin 50°.
In that triangle cos θ is adjacent over hypotenuse, adjacent over 1, which is just the adjacent side: the across. And sin θ is the opposite side: the height. So the triangle trig you know and the coordinates agree, and the point at 50° is about (0.64, 0.77).
That is why the unit circle defines sine and cosine for every angle: inside quadrant I it is triangle trig, and beyond it the coordinates carry on where triangles cannot.
Challenge
The triangle sits under the point at 35° on the unit circle: its hypotenuse is the radius 1, and x is the across. Tap the angle and the sides, name the ratio, then round x to two decimal places. Then press Check.
Try it yourself
Use the figure to find x
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Angles and the Unit Circle
1A Point on a Circle
Height on the CircleAcross: the CosineSame Height, Two AnglesRecap: Height and Across4Radians
What a Radian IsDegrees to RadiansRadians to DegreesRadians Without πTurns in RadiansRecap: Radians5Arc Length and Sector Area
Arc LengthArcs From DegreesFinding the AngleSector AreaRecap: A Slice of Pizza3Coterminal Angles
Coterminal AnglesBack Into One TurnValues Repeat Every TurnEvery Turn at OnceRecap: Coterminal Angles6The Unit Circle
Why (cos θ, sin θ)The 30° and 60° PointsThe 45° PointPoints Off the TableRecap: Quadrant I Points