From a Point to the Angle
Part of the Angles and the Unit Circle section of Coddy's Trigonometry journey. Lesson 48 of 57.
Run it backwards. If the terminal side of θ passes through a point (x, y) other than the origin, the distance to the origin is r = √(x^2 + y^2), and dividing by r takes you back to the unit circle: cos θ = x/r, sin θ = y/r, and tan θ = y/x when x is not 0.
Take (−3, 4). Then r = √(9 + 16), which is 5, so sin θ = 4/5, cos θ = −3/5 and tan θ = −4/3. The signs come out on their own, because x and y carry them.
To find θ itself, use an inverse: arccos(−3/5) ≈ 126.87°, which is in quadrant II, as the point is. A calculator's arccos always answers between 0° and 180°, and its arcsin between −90° and 90°, so check the answer against the point's quadrant.
Challenge
The terminal side of θ passes through (−5, 12). Find θ between 0° and 360°: write θ ≈ ... in degrees, rounded to two decimal places, keeping the ° sign. The calculator can work out an inverse. Then press Check.
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
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: Radians7Reference Angles and Signs
Reference AnglesReference in III and IVSigns by QuadrantSign and SizeSigns at Any AngleRecap: Reference Angles10Six Functions, Any Circle
Any Circle, Any RadiusFrom a Point to the AngleSecant, Cosecant, CotangentReciprocals in RadiansRecap: Six Functions5Arc Length and Sector Area
Arc LengthArcs From DegreesFinding the AngleSector AreaRecap: A Slice of Pizza