Trigonometry
From right-triangle ratios to generalized angles, radian measure, and the periodic sine, cosine, and tangent functions.
- In a right triangle, the hypotenuse is the longest side (opposite the right angle).
- The Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.
- The angles of any triangle add up to \(180^\circ\).
- Similar triangles have equal corresponding angles and proportional sides.
1.1 Solving problems involving triangles
In a right triangle, the trigonometric ratios relate an acute angle \(\alpha\) to the sides:
A useful identity, valid for any angle, is the fundamental relation:
A ladder \(5\ \text{m}\) long leans against a wall making a \(65^\circ\) angle with the ground. How high up the wall does it reach?
The height is the side opposite the \(65^\circ\) angle, so \(\sin 65^\circ = \dfrac{h}{5}\), giving \(h = 5\sin 65^\circ \approx 5 \times 0.9063 \approx 4.53\ \text{m}.\)
1.2 Exact ratios of 30°, 45° and 60°
These values come from the equilateral triangle (30°/60°) and the isosceles right triangle (45°), and should be memorized.
| \(\alpha\) | \(\sin\alpha\) | \(\cos\alpha\) | \(\tan\alpha\) |
|---|---|---|---|
| \(30^\circ\) | \(\tfrac{1}{2}\) | \(\tfrac{\sqrt{3}}{2}\) | \(\tfrac{\sqrt{3}}{3}\) |
| \(45^\circ\) | \(\tfrac{\sqrt{2}}{2}\) | \(\tfrac{\sqrt{2}}{2}\) | \(1\) |
| \(60^\circ\) | \(\tfrac{\sqrt{3}}{2}\) | \(\tfrac{1}{2}\) | \(\sqrt{3}\) |
1.3 Approximate values with the calculator
For angles that are not "special", we use a calculator. Always check whether it is in degree or radian mode. The inverse keys \(\sin^{-1}, \cos^{-1}, \tan^{-1}\) recover an angle from a ratio.
To find the angle whose sine is \(0.5\): \(\alpha = \sin^{-1}(0.5) = 30^\circ\). The inverse functions return a single "principal" angle, so always consider the context of the problem for other solutions.
1.4 Angles and generalized arcs; the trigonometric circle
The trigonometric (unit) circle has radius \(1\) centred at the origin. An angle is measured from the positive \(x\)-axis, positive counter-clockwise. For a point \(P=(x,y)\) on the circle at angle \(\alpha\):
A generalized angle can exceed \(360^\circ\) or be negative. Angles that differ by a whole number of full turns are coterminal and share the same trigonometric values:
\[ \sin(\alpha + k\cdot 360^\circ) = \sin\alpha, \quad k \in \mathbb{Z}. \]Signs by quadrant:
| Quadrant | \(\sin\) | \(\cos\) | \(\tan\) |
|---|---|---|---|
| I | + | + | + |
| II | + | − | − |
| III | − | − | + |
| IV | − | + | − |
1.5 Radian measure and reduction to the first quadrant
One radian is the central angle subtending an arc equal in length to the radius. A full turn is \(2\pi\) radians, so:
\[ 180^\circ = \pi \text{ rad} \qquad\Longrightarrow\qquad 1^\circ = \frac{\pi}{180}\text{ rad}, \quad 1\text{ rad} = \frac{180^\circ}{\pi} \approx 57.3^\circ. \]Common conversions: \(30^\circ = \tfrac{\pi}{6}\), \(45^\circ = \tfrac{\pi}{4}\), \(60^\circ = \tfrac{\pi}{3}\), \(90^\circ = \tfrac{\pi}{2}\).
Reduction to the first quadrant lets us evaluate any angle using a reference angle. For example:
Evaluate \(\cos\dfrac{2\pi}{3}\). Since \(\dfrac{2\pi}{3} = \pi - \dfrac{\pi}{3}\), \(\cos\dfrac{2\pi}{3} = -\cos\dfrac{\pi}{3} = -\dfrac{1}{2}.\)
1.6 Trigonometric functions and periodic phenomena
Treating the angle as a real variable \(x\) gives the functions \(f(x)=\sin x\), \(g(x)=\cos x\), \(h(x)=\tan x\).
- \(\sin x\) and \(\cos x\) have period \(2\pi\), domain \(\mathbb{R}\), and range \([-1,1]\).
- \(\tan x\) has period \(\pi\), with vertical asymptotes at \(x = \tfrac{\pi}{2} + k\pi\).
- \(\sin x\) is odd: \(\sin(-x) = -\sin x\). \(\cos x\) is even: \(\cos(-x) = \cos x\).
Many real phenomena (tides, sound, alternating current) are modelled by the general sinusoid:
where \(|a|\) is the amplitude, the period is \(\dfrac{2\pi}{|b|}\), \(c\) controls the phase shift, and \(d\) is the midline (vertical shift).
For \(f(x) = 3\sin\!\big(2x\big) + 1\): amplitude \(=3\), period \(=\tfrac{2\pi}{2}=\pi\), midline \(y=1\), so \(f\) oscillates between \(-2\) and \(4\).
Practice problems
- A right triangle has hypotenuse \(10\) and one acute angle \(30^\circ\). Find the two legs.
Show solution
Opposite the \(30^\circ\): \(10\sin 30^\circ = 10\cdot\tfrac12 = 5\). Adjacent: \(10\cos 30^\circ = 10\cdot\tfrac{\sqrt3}{2} = 5\sqrt3 \approx 8.66\). - Convert \(135^\circ\) to radians and evaluate \(\sin 135^\circ\).
Show solution
\(135^\circ = \tfrac{3\pi}{4}\). Since \(135^\circ = 180^\circ - 45^\circ\), \(\sin 135^\circ = \sin 45^\circ = \tfrac{\sqrt2}{2}\). - Given \(\cos\alpha = \tfrac{3}{5}\) with \(\alpha\) in the first quadrant, find \(\sin\alpha\) and \(\tan\alpha\).
Show solution
\(\sin^2\alpha = 1 - \tfrac{9}{25} = \tfrac{16}{25}\Rightarrow \sin\alpha = \tfrac45\) (positive in QI). \(\tan\alpha = \tfrac{4/5}{3/5} = \tfrac43\). - State the amplitude, period, and midline of \(g(x) = -2\cos\!\big(\tfrac{x}{2}\big) + 3\).
Show solution
Amplitude \(=|-2| = 2\); period \(=\tfrac{2\pi}{1/2} = 4\pi\); midline \(y = 3\).
Summary
- Right-triangle ratios: \(\sin,\cos,\tan\); fundamental identity \(\sin^2+\cos^2=1\).
- Special angles \(30^\circ,45^\circ,60^\circ\) have exact ratios worth memorizing.
- The unit circle extends the ratios to all angles; signs depend on the quadrant.
- Radians: \(180^\circ = \pi\); reduce any angle to a first-quadrant reference angle.
- Sinusoids \(a\sin(bx+c)+d\) model periodic phenomena.
- Convert \(\tfrac{5\pi}{6}\) to degrees.
- Evaluate \(\tan\tfrac{\pi}{3}\).
- Find the period of \(f(x) = \sin(3x)\).