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    • ▾Trigonometry
      • ▸Angle measures and circles
        • •Intro to radians
        • •Arc length and sector formulas
        • •Converting between radians and degrees
        • •Degrees-minutes-seconds
      • ▸Right triangles and trig
        • •Intro to trigonometry
        • •Intro to the six basic trig functions
        • •Evaluating the six basic trig functions
        • •Evaluate the remaining basic trig functions
        • •Reciprocal and quotient identities
        • •Cofunction identities
        • •Special right triangles
        • •Area of an equilateral triangle
        • •Solving right triangles
        • •Circumradii of equilateral and isosceles triangles
      • ▸Graphing trig functions
        • •Graphing sin and cos
        • •Graphing tan and cot
        • •Graphing sec and csc
      • ▸Unit circle
        • •Finding the point on the unit circle
        • •Evaluating trig functions and solving trig equations using the unit circle
        • •Evaluating trig functions using coterminal and reference angles
        • •Proofs using the unit circle
        • •Signs of trig functions
      • ▸Inverse trig functions
        • •Composition of trig functions
        • •Evaluating inverse trig functions using the unit circle
        • •Evaluating inverse trig functions using a calculator
        • •Find the inverse of a trigonometric function
        • •Graphing inverse trigonometric functions
      • ▸Non-right triangles and trig
        • •Sine rule for area of a triangle
        • •Area of a segment
        • •Heron's formula
        • •Law of cosines
        • •Parallelogram law
        • •Stewart's theorem
        • •Law of sines
        • •Bretschneider's formula, and Brahmagupta's
      • ▾Trigonometric identities
        • •Double and half angle formulas
        • ▸Solving trig equations
          • •Solving trigonometric equations by graphing
          • •Solving trigonometric equations by factoring out the GCD
          • •Finding general solutions to trig equations
          • •Solving harder trig equations by factoring
          • •Solving trigonometric equations quadratic in form
          • •t-results
        • •Solving trig equations by factoring trinomials
        • •Sum and difference formulas
        • •Auxiliary angle method
        • •Finding the acute angle between two lines
        • •Product-to-sum and sum-to-product formulas
        • •Proving inverse trig identities
        • •Power reducing trig identities
        • •Pythagorean identities
        • •Disproving trigonometric identities by counterexample
        • •Even-odd trig identities
      • ▸Polar coordinates
        • •Plotting polar points
        • •Graphing polar equations
        • •Converting between polar and Cartesian form
      • ▸Complex numbers in polar form
        • •Multiplying and dividing complex numbers in polar form
        • •Proving de Moivre's theorem
        • •Powers of complex numbers using De Moivre's theorem
        • •Roots of complex numbers using De Moivre's theorem
      • ▸Hyperbolic and inverse hyperbolic functions
        • •Proving inverse hyperbolic identities
        • •Intro to hyperbolic functions
        • •Proving hyperbolic identities
        • •Solving hyperbolic equations
     › Trigonometry › Trigonometric identities

    Sum and difference formulas

    Students will learn what the sum and difference formulas are and see several ways of proving them. Then students will learn how to use the formulas to evaluate certain expressions, solve certain equations, and prove certain identities.

    Conclude by giving your students these challenges:

    • What a Joke by NRICH
    • Tubular Stand by NRICH
    • 2020 AMC 8, Problem 4

    How Many Eggs? by NRICH

    Here's the solution: From the problem, we can deduce the following:

    $$\begin{align} & P + M + A + J = 38 \\[0.5em] & J = P - 1 \\[0.5em] & P = M - 5 \\[0.5em] & A = \dfrac{M}{2} \\[0.5em] & P = A + 2 \end{align}$$

    Next, notice that \(P\) and \(A\) are expressed in terms of \(M.\) Thus, if we could express \(J\) in terms of \(M,\) we would have all quantities in terms of \(M.\) This is easily done:

    $$J = P - 1 = (M - 5) - 1 = M - 6$$

    Next, we rewrite our very first equation in terms of \(M,\) then simplify.

    $$\begin{align} & P + M + A + J = 38 \\[0.5em] & M - 5 + M + \dfrac{M}{2} + M - 6 = 38 \\[0.5em] & 2M + \dfrac{M}{2} - 11 = 38 \\[0.5em] & \dfrac{7M}{2} - 11 = 38 \\[0.5em] & \dfrac{7M}{2} = 49 \\[0.5em] & \dfrac{M}{2} = 7 \end{align}$$

    \(A = M/2,\) and \(M/2 = 7,\) tells us \(A = 7.\) Multiplying both sides of \(M/2 = 7\) by \(2,\) we also find \(M = 14.\) Following the equations for \(J\) and \(P,\) we find \(J = M - 6 = 8,\) and \(P = M - 5 = 9.\) In summary,

    $$\begin{align} & A = 7 \\ & M = 14 \\ & J = 8 \\ & P = 9 \end{align}$$

    Is there any square with integer side lengths, such that its area is equal to its perimeter? If no such square exists, explain why. If at least one such square exists, can you find one? Can you find more than one? Can you find exactly how many exist, and explain why? What if instead of squares, we asked the same questions about rectangles?

    Here's a solution using geometry: For any rectangle with integer side lengths, having its width \(\ge 2,\) and its height \(\ge 2,\) let's consider the unit squares whose edges contribute to the perimeter of the rectangle. Notice that each corner square contributes \(2\) to the perimeter. Every non-corner square contributes \(1.\) Because a rectangle has \(4\) corners, there are \(4\) less unit squares than the perimeter of the rectangle. So to compensate, there must be exactly 4 squares inside the rectangle, which do not contribute to its perimeter. There are exactly two ways to arrange 4 squares. Namely, \(1 \times 4\) and \(2 \times 2.\) Thus, when width \(\ge 2,\) and height \(\ge 2,\) there are exactly \(2\) rectangles with integer side lengths, such that their area matches their perimeter. What if width or height is \(\lt 2?\) In this case, each of the two squares at the end contribute \(3\) to the perimeter of the rectangle, and each of the middle squares contribute \(2.\) But there is no way to compensate for this excess by adding squares to the inside of the rectangle, because such rectangles have no inside. Thus, no rectangle with integer side lengths, having width or height \(\lt 2,\) has its area equal to its perimeter.

    Here's a solution for squares, using algebra: Let \(s\) be one side of the square. Then putting together our area and perimeter formulas together, gives us \(s^2 = 4s.\) Because \(s\) is a length, we can divide both sides by \(s,\) yielding \(s = 4.\) Thus, this is the only square with its area equal to its perimeter.

    Here's a solution for rectangles, using algebra:

    $$\begin{align} & wh = 2w + 2h \\[1em] \end{align}$$

    Solving for \(w,\) then separating the rational expression into its integer and fractional parts:

    $$\begin{align} w &= \dfrac{2h}{h - 2} \\[1em] &= \dfrac{2h - 4 + 4}{h - 2} \\[1em] &= \dfrac{2(h - 2) + 4}{h - 2} \\[1em] &= 2 + \dfrac{4}{h - 2} \end{align}$$

    So \(h - 2\) must divide \(4.\) This will only be the case when \(h - 2\) is \(1,\) \(2,\) or \(4.\) When \(h = 1\) we get the same rectangle as when \(h = 4.\) Hence, this method also gives us two unique rectangles.


    TODO: Should we teach this skill of removing the variable from the numerator? Could we generalize the algebraic or geometric solution to cubes or cuboids?

    Lessons and practice problems