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The 45° Point

Part of the Angles and the Unit Circle section of Coddy's Trigonometry journey. Lesson 27 of 57.

At 45° the triangle under the point has two 45° angles, so its legs are equal. Call each a. Pythagoras with hypotenuse 1: a^2 + a^2 = 1, so a^2 = 1/2 and a = √(2)/2, about 0.71.

$The point at 45° on the unit circle, (√2/2, √2/2)$

So the point at 45° is (√(2)/2, √(2)/2): across and height are equal, which is the only place in quadrant I where that happens. The three special points of quadrant I use just three sizes: 1/2, √2/2 and √3/2.

Challenge

Tap every angle in one turn where the height is √(2)/2. 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 Circle Across: 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° PointsThe 45° PointPoints 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