Trigonometric Functions: common misconceptions

If your child says… “The turn count starts again when I pass the starting point.”

Ask together

After a full turn, has the distance travelled around the circle disappeared?

What’s true

For anticlockwise travel, arc length equals radius times radian measure, including complete turns.

If your child says… “To change degrees into radians, I multiply by 180 over pi.”

Ask together

Which multiplier changes 180 degrees into pi radians?

What’s true

One degree is a small fraction of a radian, so use pi divided by 180.

If your child says… “A half-turn brings back the same sine and cosine.”

Ask together

After half a turn from the top, is the point above or below the centre?

What’s true

A half-turn reaches the opposite point and reverses both coordinate signs.

If your child says… “One plus cotangent squared equals secant squared.”

Ask together

When dividing by sine squared, does the right side become secant squared or cosecant squared?

What’s true

Cotangent divides cosine by sine; cosecant is sine’s reciprocal, and zero denominators are excluded.

If your child says… “Sine can give any real number because every angle works.”

Ask together

Where could P sit on this circle for its height to exceed one?

What’s true

Accepting every angle does not let a coordinate extend beyond the unit circle.

If your child says… “Cosine of a sum is just the two cosines added.”

Ask together

At two zero angles, would adding their cosines still give cosine of zero?

What’s true

Squaring coordinate differences produces products, so cosine cannot be split across addition.

If your child says… “The sine version is twice sine squared minus one.”

Ask together

At zero, which order gives the cosine value one?

What’s true

The sine version subtracts twice sine squared from one.

If your child says… “The sine takes half the difference.”

Ask together

Which angle remains inside sine after the opposite terms cancel?

What’s true

In this sine sum, the middle angle belongs inside sine.