A note for the parent

What your child had to do in this Introduction to Trigonometry problem

This was not a watch-and-copy exercise. The student had to make 6 small mathematical decisions and correct any wrong turns before moving forward.

The skill being built

Reasoning, not answer recognition

Use the Pythagorean identity

\sec^2\theta - \tan^2\theta = 1

to factor into

(\sec\theta + \tan\theta)(\sec\theta - \tan\theta) = 1

. Since their sum is

3

, their difference is

\dfrac{1}{3}

. Adding and subtracting yields

\sec\theta = \dfrac{5}{3}

(so

\cos\theta = \dfrac{3}{5}

) and

\tan\theta = \dfrac{4}{3}

. Hence

\sin\theta = \tan\theta \cdot \cos\theta = \dfrac{4}{5}

The mistake we watch for

A wrong answer is useful evidence

Converting to

\dfrac{1 + \sin\theta}{\cos\theta} = 3

and squaring both sides into a complicated quadratic equation with extraneous solutions.

What happens when a step goes wrong?

The choice is crossed out

The app does not pretend the mistake did not happen.

The reason is explained

The child sees the exact misconception behind that choice.

The child tries again

The app guides, but the student still makes the next decision.

If this teaching style helps

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