Unit 5 of 10

Analytic Trigonometry and Triangles

  • 13 lessons
  • Parts 115 to 140

Once the trigonometric functions are defined, this unit works with them algebraically. It covers the fundamental identities, the inverse trigonometric functions with their principal branches, and how to prove that two trigonometric expressions are equal.

The sum, difference, double-angle, half-angle and product-to-sum formulas follow, and they are put to use solving trigonometric equations. The unit ends with the laws of sines and cosines, which solve triangles that have no right angle and give their areas.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Fundamental Identities

  1. Fundamental Trigonometric Identities

    Why sine squared plus cosine squared is always 1, how the reciprocal, quotient and even and odd identities follow from the unit circle, and why each identity holds only where both sides are defined.

    Theory part 115Practice part 116

Inverse Trigonometric Functions

  1. Inverse Sine and Cosine Functions

    How restricting sine and cosine to one branch makes them invertible, which angle arcsine and arccosine return, and why they only accept inputs that a sine or cosine can actually produce.

    Theory part 117Practice part 118

  2. Inverse Tangent and Reciprocal Trigonometric Functions

    Why arctangent accepts every real number and always returns an angle smaller than a right angle in size, and how inverse cotangent, secant and cosecant depend on a stated choice of branch.

    Theory part 119Practice part 120

  3. Compositions Involving Inverse Trig Functions

    How to evaluate a trig function of an inverse trig value exactly with a reference triangle, why undoing in the other order can fail, and how principal ranges fix every sign.

    Theory part 121Practice part 122

Identities and Formulas

  1. Verifying Trigonometric Identities

    How to prove that two trigonometric expressions agree wherever both are defined, by rewriting one side with factoring, common denominators and conjugates, and why testing values never proves it.

    Theory part 123Practice part 124

  2. Sum and Difference Formulas

    How to find the sine, cosine and tangent of a sum or difference of two angles from the values of each angle, why the formulas hold, and how they give exact values such as sin 75°.

    Theory part 125Practice part 126

  3. Double-Angle Formulas

    How the sum formulas with two equal angles give sin 2x, the three forms of cos 2x and tan 2x, how to pick the right form, and how to find all of them from one value and a quadrant.

    Theory part 127Practice part 128

  4. Power-Reducing and Half-Angle Formulas

    How to trade a square of sine or cosine for a cosine of twice the angle, and how to find exact values at half an angle with the correct sign.

    Theory part 129Practice part 130

  5. Product-to-Sum and Sum-to-Product Formulas

    How the angle addition formulas turn a product of sines and cosines into a sum and a sum back into a product, and why two nearly equal tones produce a slow beat.

    Theory part 131Practice part 132

Trigonometric Equations

  1. Solving Linear Trigonometric Equations

    How to isolate a sine, cosine or tangent, get one solution from an inverse function, and then write every solution, either as a periodic family or as a list in a given interval.

    Theory part 133Practice part 134

  2. Solving Quadratic Trigonometric Equations

    How to solve trigonometric equations that factor or become quadratics in one function, how to handle multiple angles over the right interval, and how to reject candidates that fail the original equation.

    Theory part 135Practice part 136

Solving Triangles

  1. The Law of Sines and the Ambiguous Case

    How the law of sines solves triangles that have no right angle, and why two sides with an angle that is not between them can give zero, one or two triangles.

    Theory part 137Practice part 138

  2. The Law of Cosines and Triangle Area

    How the law of cosines finds a side from two sides and the angle between them, or an angle from three sides, and how to get the area of a triangle from its sides and angles.

    Theory part 139Practice part 140