Height on the Circle

Part of the Angles and the Unit Circle section of Coddy's Trigonometry journey. Lesson 1 of 55.

A calculator says $\sin\left(135^{\circ}\right)$ is 0.7071. But no right triangle has a 135° angle. So what is a sine past 90°?

Turn a point counterclockwise from (1, 0) around a circle of radius 1. The hypotenuse is always the radius, 1. So opposite divided by hypotenuse is the point's height. That height is $\sin\left(\theta\right)$ , at every angle.

$The point P at 135° on the circle of radius 1, with its height dashed down to the x-axis$

Challenge

Tap every angle in one turn where the point is half a unit above the x-axis. That is where $\sin\left(\theta\right) = \frac{1}{2}$ .

Then press Check.

Start the lesson to read it in the interactive player and track your progress.

All lessons in Angles and the Unit Circle

1A Point on a Circle

Height on the CircleAcross: the Cosine Same Height, Two Angles Recap: Height and Across

4Radians

What a Radian Is Degrees to Radians Radians to Degrees Radians Without π Turns in Radians Recap: Radians

7Reference Angles and Signs

Reference Angles Reference in III and IV Signs by Quadrant Sign and Size Signs at Any Angle Recap: Reference Angles

10Six Functions, Any Circle

Any Circle, Any Radius From a Point to the Angle Secant, Cosecant, Cotangent Reciprocals in Radians Recap: Six Functions

2Standard Position

Angles in Standard Position Negative Angles Recap: Placing Angles

5Arc Length and Sector Area

Arc Length Arcs From Degrees Finding the Angle Sector Area Recap: A Slice of Pizza

8All Four Quadrants

Points in Quadrant II Quadrants III and IV Every Angle for a Value Points in Radians Exact Values in Radians Recap: Four Quadrants

11Final Challenges

The Minute Hand The Ferris Wheel Seat Four Angles, Two Turns The Garden Sprinkler A Negative Radian Point Which Angle?

3Coterminal Angles

Coterminal Angles Values Repeat Every Turn Every Turn at Once Recap: Coterminal Angles

6The Unit Circle

Why (cos θ, sin θ)The 30° and 60° Points The 45° Point Points Off the Table Recap: Quadrant I Points

9Tangent on the Circle

Tangent Is y over x Where Tangent Is Undefined Tangent in Radians Every Angle for a Tangent Recap: Tangent