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Height on the Circle

Part of the Angles and the Unit Circle section of Coddy's Trigonometry journey. Lesson 1 of 57.

Take a circle of radius 1 centred at the origin. Put a point at its right-hand end, (1, 0), and turn it counterclockwise round the circle. The angle it has turned through is called θ, and the point's height above the x-axis has a name: the sine of θ, written sin(θ).

$The circle of radius 1 with a point P turned through θ; a dashed line drops from P to the x-axis$

The dashed line is the height of P: that is sin θ.

Follow the height through one turn. It climbs from 0 to its largest value at the top, a quarter turn round: sin(90°) = 1. The point cannot rise higher, because the circle reaches only one radius above the centre. After the top the height falls, goes negative once the point is below the x-axis, and bottoms out at sin(270°) = -1 before climbing back. So sin θ is always between -1 and 1.

Challenge

Tap every angle in one turn where the point's height is 0: the angles with sin(θ) = 0. Then press Check.

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

Test yourself

This 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 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 Quadrants and Axes 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 Back Into One Turn 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