MATHS 460 - BALL STATE UNIVERSITY - SUMMER 2000

The Role of Algebra - Bridging the Gaps

In trigonometry today, we use algebra constantly in order to solve equations indirectly, prove identities, and solve right triangles.  Here's a couple of examples.

Example 1

We have used the sine and cosine functions to find lengths of the sides of right triangles if we knew the angles.  Now we can use algebra in the example below - we have to find the hypotenuse length in order to use our functions.

Find the sine and cosine of the angle θ.

In order to use our functions, we have to know the length of the hypotenuse of the triangle in the figure at the left.  In order to compute this length, we have to use the Pythagorean Theorem.  Therefore, 6² + 8² = x².  Simplifying this result gives us 36 + 64 = x², or 100 = x².  Therefore x = √100, and we find x to be 10 units in length.  Therefore, sin θ = 6/10 or 3/5, and cos θ = 8/10 or 4/5.

Example 2

We can also use algebra to prove some trigonometric identities and simplify expressions.  For example:

Simplify:   sin θ sec θ cot θ

Before we start, it is important to introduce four other trigonometric functions:  tangent, secant, cosecant, and cotangent.  The tangent function, tan θ, is equal to the ratio of the length of the side of the triangle opposite the angle θ, divided by the length of the side of the triangle adjacent the angle θ.  Furthermore, it can be proved that tan θ = sin θ/cos θ.  The remaining three functions are called reciprocal functions.  In short:  sec θ = 1/cos θ, csc θ = 1/sin θ, and cot θ = 1/tan θ, or cos θ/sin θ.  Now we can simplify our expression.

Since sec θ = 1/cos θ and cot θ = cos θ/ sin θ, we then would have (sin θ) X (1/cos θ) X (cos θ/ sin θ).  This simplifies to (sin θ cos θ) / (cos θ sin θ)  Since our numerator in our fraction equals the denominator, this expression reduces to 1.  However, without algebra, we wouldn't know were to go or what to do next.

We can even now turn back to Ptolemy's ancient mathematics, introduce the concepts of algebra, and derive Ptolemy's famous theorems be working backwards through the steps of his proofs.  With algebra, we can analyze Ptolemy's mathematics even further, and in return, understand it better. 

Why Use Trigonometry?

Trigonometry is perhaps one of the best measurement tools available for us to use.  If we know basic angle measures, we can fin distances.  Furthermore, if we know distances, we can find angle measures.  With trigonometry, we can compute the height of buildings, angles of road curves, and areas and volumes of various shapes and figures.  But the uses for trigonometry don't stop there.  There are infinitely many possibilities to use trigonometry, which is why it is so important for students to learn about this amazing branch of mathematics during their high school careers.  

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