Complete, Concrete, Concise » Mathematics
Understanding Supplementary and Complementary Angles
Supplementary and complementary angles are useful concepts to understand in trigonometry. Supplementary angles are any two angles that sum (add) up to 180° (a straight line). An easy way to remember this is that Supplementary and Straight line both begin with the letter ‘S’. Complementary angles are any two angles that sum (add) up to 90° (a corner). An easy way to remember this is that Complementary and Corner both begin with the letter ‘C’. In the above image, we see that angles a and b sum up to 90°, therefore they are complementary angles. In the above image, we see that angles c and d sum up to 180°, therefore they are supplementary angles. While the angles were drawn adjacent to one another, they don’t have to be – although, that is the most common way … Read entire article »
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Proving the Law of Sines
A tutorial on the Law of Sines can be found here. Assumes an understanding of the trigonometric function sine. A basic tutorial can be found here and a tutorial using the Unit Circle can be found here. The Law of Sines says that “given any triangle (not just a right angle triangle): if you divide the sine of any angle, by the length of the side opposite that angle, the result is the same regardless of which angle you choose”. The actual value (the result of the calculation) is equal to the diameter of the smallest circle you can draw around the triangle that has all three points of the triangle touching the edge of the circle (a circumscribed circle). We could state the Law of Sines more formally as: for any triangle, the … Read entire article »
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Understanding the Law of Sines
While trigonometry is used to solve problems involving right angle triangles, it can also be applied to triangles that are not right angle triangles. Assumes you understand basic trigonometric concepts. A tutorial on understanding sine, cosine and tangent can be found here. A tutorial on understanding the trigonometric functions and the unit circle can be found here. What is the Law of Sines One of the fundamental properties of two triangles that have the same shape (i.e., they have the same angles) is that the ratio of any two sides is identical – regardless of the size of the triangles. a A a A b B — = — and — … Read entire article »
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Understanding Basic Trigonometric Identities
This tutorial assumes you are familiar with the trigonometric functions and their derivation from the unit circle. A tutorial on the trigonometric functions can be found here. A tutorial on the trigonometric functions and the unit circle can be found here. What is an Identity? An identity (in mathematics) is something that is true (more precisely: a tautological relationship). For example, the following is an identity: 1 + 1 = 2 It says that 1 + 1 is exactly the same as 2. A common (and important) trigonometric identity is: sin2θ + cos2θ = 1 All this identity says is that what is on the left hand side is identical to what is on the right hand side. Why Learn / Use Identities? Identities (in any branch of mathematics) help us to: solve simplify or gain insight into mathematical problems. Identities are a lot like synonyms in … Read entire article »
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How to Solve Trigonometric Problems
This tutorial offers advice on how to solve trigonometric problems and provides several problems worked through in detail. It assumes you are familiar with the trigonometric functions sine, cosine, tangent, secant, cosecant, and cotangent. A basic tutorial can be found here. A more advanced tutorial can be found here. If an explanation / walkthrough is not clear, please let me know in a comment and I will try to improve the answer. Identify what it is that is not clear – just saying “I didn’t understand it” isn’t very helpful. If you have a problem you would like to see solved, leave a comment. Note: (1) I don’t do homework, (2) I don’t promise I will solve it, (3) I don’t promise to solve it quickly. Don’t Panic Don’t to panic or freak out. No one … Read entire article »
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Understanding Trigonometric Functions using the Unit Circle (Advanced)
While I call this advanced, it does not mean harder or more complicated, it just means more abstract. Understanding the trigonometric functions (sine, cosine, tangent) using right angle triangles is simply a special case of trigonometric functions using the unit circle. I strongly recommend first reading and understanding the article Understanding Sine, Cosine, and Tangent first, because it explains the history and reasoning behind the trigonometric functions. This is the way trigonometric functions are generally understood and defined in mathematics. Trigonometric functions were originally developed and understood from the study of right angle triangles. The problem with using right angle triangles is that trigonometric functions can only be defined for angles between 0° and 90°, but not for angles ≤ 0° or ≥90° because no such right angle triangles exist. The Unit Circle The Unit … Read entire article »
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Understanding Sine, Cosine and Tangent
An article explaining trigonometric functions using the unit circle can be found here Using the unit circle is the standard way trigonometric functions are defined and understood in mathematics. I recommend reading and understanding this article first. Later, if you want to understand how trigonometric functions are defined for values greater than 90° or less than 0°, go and read the other article. Sine is often introduced as follows: Which is accurate, but causes most people’s eyes to glaze over. The problem is that from the time humans starting studying triangles until the time humans developed the concept of trigonometric functions (sine, cosine, tangent, secant, cosecant and cotangent) was over 3000 years. A Little History The ancients studied triangles. One of the things they did was to compare the lengths of the sides of triangles: A triangle has … Read entire article »
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Understanding Averages – Mean, Median, and Mode
This tutorial examines the concept of the average as a single value representing a collection of values. It focusses on the mean, median, and mode. The average (especially in physics) can also mean the center or balance point, but, for most everyday use, we tend to think of the average as representative value. Average comes from the Old French avarie which came from the Old Italian avaria which came from the Arabic awariyah meaning damaged goods or merchandise. Which is probably apt, given how how badly averages are often misused. The Old French avarie used to mean the damage sustained to a ship or its cargo. The meaning later shifted to mean an equal distribution of the costs of such damage. For example, if ten men pooled money together and hired a ship … Read entire article »
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Proving Alternate Interior Angles are Congruent (the same)
The Alternate Interior Angles Theorem states that If two parallel straight lines are intersected by a third straight line (transversal), then the angles inside (between) the parallel lines, on opposite sides of the transversal are congruent (identical). This is illustrated in the image below: We see two parallel lines and a third line (transversal) intersecting (crossing or cutting through) both of them. The green shaded angles are: (1) inside (between) the two parallel lines, (2) congruent (identical or the same), and (3) on opposite sides of the transversal. This is true for the other two unshaded interior angles. It is also true for the alternate exterior angles (but not proved here). Axioms Proofs are built on two things: (1) postulates, axioms, or hypotheses – these are things that are assumed to be true, but can’t … Read entire article »
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Proving that the Angles in a Triangle Sum up to 180°
Proving that the angles inside a triangle – any triangle – sum up to 180° is very simple, but leaves most people unsatisfied (or unconvinced) because it depends on the properties of something called Alternate Interior Angles. I’ll give the proof first and then explain Alternate Interior Angles A demonstration of the angles of a triangle summing up to 180° can be found here. Proof Given any triangle, how can you prove that the angles inside a triangle sum up to 180°? 1) Draw a line parallel to one of the sides of the triangle that passes through the corner opposite to that side: It is easiest to draw the triangle with one edge parallel to the horizontal axis, but you don’t have to because this proof works regardless of the orientation of the triangle. Rotating … Read entire article »
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