Derive the Trigonometric Identities

This examples shows how to derive the trigonometric identities using algebra and the definitions of the trigonometric functions. The identities can also be derived using the geometry of the unit circle or the complex plane[1][2]. The identities that this example derives are summarized below:

  1. Derive Pythagorean Identity
  2. Derive Sum of Two Angles Identities
  3. Derive Difference of Two Angles Identities
  4. Derive Double Angle Identities (Algebra)
  5. Derive Half Angle Identities (Algebra)

Derive Pythagorean Identity

Pythagorean Identity

To derive the pythagorean identity the lengths of the adjacent and opposite side of the right triangle are defined in terms of the cosine and sine of the angle of the right-triangle. Then, the lengths are substituted into Pythagorean’s Theorem .

  1. Lets start with the triangle formed by the unit circle, which visualizes all right triangles of hypotenuse one. Note, the adjacent side corresponds to the x-component of the right triangle and the opposite side corresponds to the y-component of the right triangle. The two components form the (x,y) point along the circumference of the circle.

    Unit Circle Notation

    Then, using the definitions of the trigonometric functions cosine and sine, we can substitute the variables which represent the adjacent side (x), the opposite side (y), and hypotenuse (1) of the right triangle into the equations.

    After simplifying the equations, the adjacent side corresponds directly with the cosine function and the opposite side corresponds with the sine function for a given angle.

  2. Next, recall the equation for Pythagorean’s Theorem which relates the squares of the sides together as shown below:

    Pythagorean Theorem

    After substituting the corresponding variables to convert the theorem into the Cartesian Coordinate System we are left with a familiar equation, the equation of a circle.

    Then, by substituting the corresponding sine and cosine function above, which we found to correspond to the x and y components of the triangle, we get Pythagorean’s identity.

Derive Sum of Two Angles Identities

This example derives the sum of two angles identities using the right triangle definitions of the functions sine and cosine. The right triangle definitions of sine and cosine are shown below.

Steps

  1. Start by drawing a right triangle with an angle and hypotenuse of as shown below. The geometry of this triangle will be used to derive the identities.

    sum of two angles identities step 1 a

    Solve for the lengths of the adjacent and opposite sides by substituting , and into the definitions of sine and cosine.

    Label these lengths in the figure.

    derive sum of two angles identities step 1 b

    Draw a line parallel to and use the corresponding angle theorem to label the corresponding angles and .

    derive sum of two angles identities step 1 c

    Draw a right triangle with the angle starting at the point that shares the hypotenuse . Then draw two more right triangles that complete the rectangular shape.

    derive sum of two angles identities step 1 d

    The geometry of this shape can be used to represent the lengths of the adjacent and opposite sides of the original right triangle.

    Substitute the sine and cosine of the angle from above. This gives us the general form of the identities, next we will find the unknown lengths.

  2. Find the adjacent and opposite lengths of the right triangle .

    derive sum of two angles identities step 2 a

    Substitute the adjacent side , opposite side and hypotenuse into the definitions of sine and cosine and solve for the adjacent and opposite sides.

    Label these lengths in the figure.

    derive sum of two angles identities step 2 b

  3. Find the adjacent and opposite lengths of the right triangle .

    derive sum of two angles identities step 3 a

    Substitute the adjacent side , opposite side and hypotenuse into the definitions of sine and cosine and solve for the adjacent and opposite sides.

    Label these lengths in the figure.

    derive sum of two angles identities step 3 b

  4. Observe that the angle is equal to , because it is complementary to the angle which is complementary to .

    derive sum of two angles identities step 4 c

    Find the adjacent and opposite lengths of the right triangle .

    derive sum of two angles identities step 4 c

    Substitute the adjacent side , opposite side and hypotenuse into the definitions of sine and cosine and solve for the adjacent and opposite sides.

    Label these lengths in the figure.

    derive sum of two angles identities step 4 c

    Substitue the unknown lengths into the equation from the end of step 1.

    This gives us the sum of two angles identities.

Derive Difference of Two Angles Identities

This example shows how to derive the difference of two angles identities using the right triangle definitions of the functions sine and cosine. The definitions are shown below.

Steps

  1. Start by drawing a right triangle with an angle of and a hypotenuse of . The angles (alpha) and (beta) are also drawn.

    derive difference of two angles identities step 1 a

    Solve for the lengths of the adjacent and opposite side. Substitute the hypotenuse , adjacent side and opposite side into the defintions of sine and cosine.

    Label these lengths in the figure.

    derive difference of two angles identities step 1 b

    The goal is to represent these side-lengths in terms of the sine and cosine of the angles and . To achieve this goal, draw another right triangle of hypotenuse with the angle of on top of the first right-triangle.

    derive difference of two angles identities step 1 b

    This allows us to represent the length of as the sum of the lengths and . And to represent the length as the difference of the lengths and .

    Substituting the expressions from above gives us the starting point for the identities. In the next steps we will solve for the unknown lengths using the definitions of sine and cosine.

  2. Solve for the adjacent and opposite sides of the right-triangle illustrated below.

    derive difference of two angles identitites step 2 a

    Substitute the hypotenuse , adjacent side and opposite side into the definitions of sine and cosine, then solve for the adjacent and opposite side.

    Label the side lengths in the figure.

    derive difference of two angles identitites step 2 b

  3. Solve for the adjacent and opposite sides of the right-triangle illustrated below. From the corresponding angle theorem we know that is the same as .

    derive difference of two angles identitites step 3 a

    Substitute the hypotenuse , adjacent side and opposite side into the definitions of sine and cosine, then solve for the adjacent and opposite side.

    Label these lengths in the figure.

    derive difference of two angles identitites step 3 b

  4. Solve for the adjacent and opposite sides of the right-triangle illustrated below. We know that is equal , because it is complementary to which is complementary angle to .

    derive difference of two angles identitites step 4 a

    Substitute the hypotenuse , adjacent side and opposite side into the defintions of sine and cosine and solve for the adjacent and opposite side.

    Label these lengths into the illustration below.

    derive difference of two angles identitites step 4 a

    This gives us all the unknown lengths in the figure.

    derive difference of two angles identitites step 4 c

  5. Substitute the lengths into the equation from the end of step 1.

    This give us the difference of two angles identities.

Derive Double Angle Identities (Algebra)

This example derives the double angle identities using algebra and the sum of two angles identities.

Steps

  1. Start with the sum of two angles identities.

  2. Substitue and into the identities. This is same as saying the angle (alpha) is equal to (beta).

  3. Combine the arguments on the left and simplify the expressions on the right. This gives us the double angle identities.

  4. Optionally, the pythagorean identity, shown below, can be used to two double angle identity variations.

    Subtract from both sides.

    Substitute this expression into the identity from step 3 and combine like terms. This gives us the first variant.

    The second variant is found by subtracting from both sides of the pythagorean identity.

    Substitute this expression into the identity from step 3 and combine like terms. This gives us the second variant.

Derive Half Angle Identities (Algebra)

This example derives the half angle identities using algebra and the double angles identities.

Steps

  1. Start with the double angle identities.

  2. Transform the equations by substituting the half-angle of (alpha) in for (theta).

  3. Simplify the left side of the equations.

  4. Add to both sides of the first equation in step 3. Then subtract from both sides.

    Divide both sides by two.

    Take the square root of both sides.

    This gives us the first half-angle identity.

  5. Rearrange the second equation from step 3 so that the half angle is on the left side.

    Add to both sides.

    Divide both sides by two.

    Take the square root of both sides.

    This gives us the second half-angle identity.

  6. In conclusion, the two half angle idenities are given below.

Links

References

  1. Derive the Trigonometric Identities (Unit Circle)
    Wumbo (internal)
  2. Derive the Trigonometric Identities (Complex Plane)
    Wumbo (internal)