Angles, the Unit Circle and Trigonometric Graphs: Quick Revision

A brief note on Angles, the Unit Circle and Trigonometric Graphs. Read it before a test. Try each question at the end before you open its answer.

Angles and the unit circle

• $360^\circ = 2\pi$ rad. Degrees to radians: multiply by $\dfrac{\pi}{180}$, or call math.radians. Python's sin, cos and tan take radians.
• The point at angle $\theta$ on the unit circle is $(\cos\theta, \sin\theta)$. Quadrant I has both positive, II has only the sine positive, III has neither, IV has only the cosine positive.
• Compare floats with math.isclose(a, b, abs_tol=...). math.sin(math.pi) is about $10^{-16}$, not $0$.

Graphs

• np.linspace(0, 2 * np.pi, n) makes the angles. np.sin works on the whole array.
• $y = A\sin\big(B(x - C)\big)$: amplitude $A$, period $\dfrac{2\pi}{B}$, shifted right by $C$. $\cos x = \sin\left(x + \dfrac{\pi}{2}\right)$.
• The tangent repeats every $\pi$ and has asymptotes. Replace the huge values with np.nan before plotting.

Identities

• One counterexample proves an identity false. Many passes make it very likely true.
• np.allclose(left, right) tests whole arrays. np.max(np.abs(left - right)) is the worst miss, about $10^{-16}$ for a true identity.
• Use random angles from rng.uniform(0, 2 * np.pi, n), and skip the angles where $\cos a = 0$ for the tangent.

A worked example

What does this print?

1. Radians. $150 \times \dfrac{\pi}{180} = 2.618$.
2. Sine. $150^\circ$ is in quadrant II, where the sine is positive. $\sin 150^\circ = \sin 30^\circ = 0.5$, printed as 0.5.
3. Tolerance. The cosine of $\dfrac{\pi}{2}$ is about $6 \times 10^{-17}$, which is close to $0$ within $10^{-9}$: True.

It prints 2.618, 0.5 and True.

Practice with new numbers

A program to try

The program checks $\sin^2 a + \cos^2 a = 1$ on four angles and compares each total with $1$ using a tolerance.

Mistakes to avoid

• Degrees given to a function that wants radians. Convert with math.radians or np.radians.
• Typing 3.14 for $\pi$. Use math.pi.
• Comparing floats with ==. Use isclose or allclose.
• A tangent plot with steep joining lines. Hide the huge values with np.nan.
• Taking a pass as a proof. Many passing angles do not prove an identity.

Check yourself

1. Convert $90^\circ$ to radians.

2. What is the period of $y = \sin(4x)$?

3. In which quadrant are both the cosine and the sine negative?

4. What does np.argmax(bad) give when bad marks the angles that fail?