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05 · Subject library
Trigonometry
Angles, triangles, periodic motion, identities, and the unit circle as one connected system.
Start with Amplitude and Period →29 complete · 29 total lessons
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01Amplitude and PeriodAmplitude measures a sinusoid's vertical departure from its midline, while period measures the input length of one complete cycle.Full lesson + sheet02Angles in Standard PositionAn angle in standard position begins on the positive x-axis and records rotation to a terminal arm.Full lesson + sheet03Area of an Oblique TriangleAn oblique triangle has no right angle. Its area can be found from two sides and their included angle: $A=\frac12ab\sin C$, where $C$ lies between sides $a$ and $b$.Full lesson + sheet04Complex Numbers in Polar FormPolar form represents a nonzero complex number by modulus and argument: $z=r(\cos\theta+i\sin\theta)$, often written $r\operatorname{cis}\theta$ or $re^{i\theta}$.Full lesson + sheet05CosineCosine measures the ratio of the side adjacent to an angle to the hypotenuse in a right triangle.Full lesson + sheet06Coterminal AnglesCoterminal angles share the same initial and terminal sides in standard position. They differ by whole rotations: in degrees $\theta+360^\circ k$, and in radians $\theta+2\pi k$, where $k$ is any integer.Full lesson + sheet07Degrees and RadiansDegrees and radians are two units for angle measure. A full turn is $360^\circ=2\pi$ radians, so $180^\circ=\pi$ radians.Full lesson + sheet08Double Angle IdentitiesDouble-angle identities rewrite trigonometric functions of 2θ in terms of functions of θ.Full lesson + sheet09Exact Trigonometric ValuesSpecial triangles, reference angles, and quadrant signs produce exact sine, cosine, and tangent values without decimals.Full lesson + sheet10Graph of TangentThe tangent function is $\tan x=\sin x/\cos x$. Its graph has period $\pi$, zeros where $\sin x=0$, and vertical asymptotes where $\cos x=0$.Full lesson + sheet11Graphs of Sine and CosineSine and cosine graphs turn circular motion into smooth repeating waves described by amplitude, period, midline, and phase.Full lesson + sheet12Half Angle IdentitiesHalf-angle identities express trig values at $\theta/2$ using functions of $\theta$: $\sin(\theta/2)=\pm\sqrt{(1-\cos\theta)/2}$, $\cos(\theta/2)=\pm\sqrt{(1+\cos\theta)/2}$, and $\tan(\theta/2)=\sin\theta/(1+\cos\theta)=(1-\cos\theta)/\sin\theta$ where denominators are nonzero.Full lesson + sheet13Law of CosinesThe law of cosines generalizes the Pythagorean theorem to any triangle and solves SAS or SSS information.Full lesson + sheet14Law of SinesThe law of sines relates every side of a triangle to the sine of its opposite angle.Full lesson + sheet15Phase ShiftPhase shift is horizontal displacement of a periodic graph. In $y=a\sin(b(x-h))+k$ or cosine form, the phase shift is $h$: right if $h>0$, left if $h<0$.Full lesson + sheet16Polar CoordinatesPolar coordinates locate a plane point by distance and direction: $(r,\theta)$ means move signed distance $r$ along the ray at angle $\theta$.Full lesson + sheet17Pythagorean IdentitiesThe fundamental identity $\sin^2\theta+\cos^2\theta=1$ comes from the unit circle equation $x^2+y^2=1$.Full lesson + sheet18Reciprocal Trigonometric FunctionsCosecant and secant are reciprocals of sine and cosine: $\csc\theta=1/\sin\theta$ where $\sin\theta\ne0$, and $\sec\theta=1/\cos\theta$ where $\cos\theta\ne0$.Full lesson + sheet19Reference AnglesA reference angle is the acute positive angle between an angle’s terminal side and the $x$-axis. It captures the magnitude of sine, cosine, and tangent values; the quadrant supplies their signs.Full lesson + sheet20Right Triangle TrigonometryRight triangle trigonometry connects an acute angle to the side-length ratios sine, cosine, and tangent.Full lesson + sheet21SineSine measures the ratio of the side opposite an angle to the hypotenuse in a right triangle.Full lesson + sheet22Sinusoidal FunctionsSinusoidal functions model smooth repeating behaviour using amplitude, midline, period, and phase shift.Full lesson + sheet23Solving Trigonometric EquationsTrigonometric equations combine algebra, reference angles, periodicity, and domain restrictions to find every valid angle.Full lesson + sheet24Sum and Difference IdentitiesSum and difference identities express trig values of combined angles using the values of the individual angles.Full lesson + sheet25TangentTangent measures the ratio of vertical change to horizontal change: opposite divided by adjacent.Full lesson + sheet26Trigonometric IdentitiesTrigonometric identities are equations true for every input where both sides are defined and can simplify, verify, or transform expressions.Full lesson + sheet27TrigonometryTrigonometry connects angles, side ratios, circles, waves, vectors, and geometric measurement.Full lesson + sheet28Unit CircleThe unit circle turns every angle into a point whose coordinates are cosine and sine.Full lesson + sheet29Vectors in Two DimensionsTwo-dimensional vectors represent magnitude and direction through components, geometric arrows, and linear combinations.Full lesson + sheet
