Trigonometry course

1. 1.Right-Triangle Trigonometry

   In a right triangle, each acute angle defines three ratios of sides. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse and tangent is opposite over adjacent, often remembered as SOH-CAH-TOA. Because similar triangles have equal ratios, these values depend only on the angle.

   • Label sides relative to the angle you are using; opposite and adjacent swap for the other acute angle.
   • Angle of elevation is measured up from the horizontal; angle of depression down from it.
   • Special triangles: $30^\circ$-$60^\circ$-$90^\circ$ sides $1 : \sqrt{3} : 2$; $45^\circ$-$45^\circ$-$90^\circ$ sides $1 : 1 : \sqrt{2}$.
   Quick check: In a right triangle the hypotenuse is 10 and an angle is $30^\circ$. Find the opposite side.

   $5$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study right-triangle trigonometry

2. 2.The Unit Circle

   The unit circle has radius 1 and centre at the origin. For an angle $\theta$ measured counterclockwise from the positive $x$-axis, the point where its terminal side meets the circle is $(\cos\theta, \sin\theta)$. This definition extends sine and cosine to every angle, including obtuse, negative and very large ones.

   • $180^\circ = \pi$ radians; multiply degrees by $\frac{\pi}{180}$ to convert.
   • Signs: all positive in quadrant I, sine in II, tangent in III, cosine in IV.
   • Reference angle: the acute angle to the $x$-axis, which fixes the absolute values.
   Quick check: Convert $135^\circ$ to radians as a decimal.

   $\frac{3\pi}{4} \approx 2.356$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study the unit circle

3. 3.Trigonometric Identities

   A trigonometric identity is an equation that holds for every angle where both sides are defined. The Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ comes straight from the unit circle, and dividing it by $\cos^2\theta$ or $\sin^2\theta$ gives two more: $1 + \tan^2\theta = \sec^2\theta$ and $1 + \cot^2\theta = \csc^2\theta$.

   • Rewrite everything in sine and cosine when stuck.
   • $\cos 2\theta$ has three forms: $\cos^2\theta - \sin^2\theta$, $2\cos^2\theta - 1$, $1 - 2\sin^2\theta$.
   • Factor and use common denominators exactly as in algebra.
   Quick check: If $\sin\theta = \frac{3}{5}$ with $\theta$ acute, find $\cos\theta$.

   $\frac{4}{5}$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study trigonometric identities

These topics belong to the course outline. Full explainers for them have not been written yet, so they are summarised here rather than given pages of their own.

Law of Sines and Cosines

Solve any triangle from three known parts.

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