Calling all math enthusiasts -- you better get ready, as we're about to dive headfirst into the captivating realm of trigonometry. Don't worry at the beginning, as it won't involve any form of complex calculations or rocket science. It would just have some cool tricks to understand triangles better. We'll be specifically taking up the case of equilateral triangles to comprehend trigonometric ratios of 30° and 60° -- the ones with all three sides and angles equal.
Looking at Trigonometric Ratios of 30° and 60°
To start with: imagine the perfectly shaped equilateral triangle, more like a slice of pizza (minus the yummy toppings!). Let's call it triangle ABC wherein all three sides, AB, BC, and AC have exactly the same length, say 2x units (just to make understanding simple: think of each unit as the length of a finger).
Additionally, let all three angles; namely, ∠A, ∠B, and ∠C have the same measure -- 60 degrees (this can be understood by thinking of a minute hand pointing to the 12 o'clock position and the hour hand pointing at 2 -- a rough assumption for understanding purpose).
Now, within the equilateral triangle, draw a line straight down going from point A, moving perpendicular to the bottom side BC, and meeting it at point D, as shown in the figure. This line splits the triangle into two smaller right-angled triangles; namely, ΔABD and ΔACD.
Here's the important part: since we started with a special equilateral triangle and broke it up into two right triangles of the same size, they both are congruent -- meaning they have exactly the same type of shape and amount of area contained within them.
Now, owing to this, we have the sides opposite to the right angles having the same length, and the angles next to the right angles (that are, ∠BAD and ∠CAD) are equal too. So, ideally, this would make both ∠BAD and ∠CAD to have the same measure of 30 degrees.
Therefore Δ ABD ≅ Δ ACD
AD^2 + BD^2 = AB^2 ….(Using the concept of Pythagoras Theorem)
AD^2 = AB^2 – BD^2
= (2x)^2 – (x)^2
= (3x)^2
AD = x√3
Sine (sin): It compares the opposite side (BD) to the hypotenuse (AB), and tells you, as a proportion, how tall that opposite side is compared to the entire diagonal of the right triangle (the hypotenuse) -- in this case: sin 30° is 1/2.
Cosine (cos): This compares the adjacent side (AD) to the hypotenuse (AB), just as a proportion, how tall that adjacent side is compared to the entire diagonal of the right triangle (the hypotenuse) -- in this case: cos 30° is √3/2.
Tangent (tan): This one connects the opposite side (BD) to the adjacent side (AD), like comparing the height of your slice to its length -- in this case, for our triangle, tan 30° is 1/√3.
So, these ratios act like unique fingerprints, revealing the characteristics of our special 30-degree triangle, just like your fingerprints identify you!
OR, if you still haven’t understood and want in a straightforward language:
Now, let us imagine a 30-degree slice of pie: its trigonometric ratios; namely, sine and cosine, when found help us to understand how angles relate to side lengths in triangles -- just like a recipe uses measurements while making any dish!
This option uses a relatable analogy (pie slice) to explain the trigonometric ratio concept and connect it to everyday life (recipe).
With this, we can write: Sin 30 degree = BD (Adjacent)/AB (Hypotenuse) = x/2x = 1/2;
Cos 30 Degree = AD (Opposite)/AB (Hypotenuse) = x√3/2x = √3/2;
Tan 30 Degree = BD (Adjacent)/AD (Opposite) = x/x√3 = 1/√3
Once Sin, Cos, and Tan have been found, it becomes easier to find out:
Cosec 30 Degree = 1/Sin 30 = 1/(1/2) = 2
Sec 30 Degree = 1/Cos 30 = 1/(√3/2) = 2/√3
Cot 30 Degree = 1/Tan 30 = 1/(1/√3) =√3
Now, moving forward, unlocking the secrets of 60 degrees, we can go on to understand the values hidden within its trigonometric ratios of sin, cos, tan, cot, cosec, and sec.
Here's the scoop:
Sine 60 -- the ratio of the opposite side to the hypotenuse -- is a neat √3/2;
Cosine 60 -- the ratio of the adjacent side to the hypotenuse -- is a cool 1/2;
Tangent 60 -- the ratio of opposite to adjacent, hits the mark at √3.
Now, flip the fractions, and you get cosecant, secant, and cotangent, revealing their values too!
Cosec 60 Degree = 1/Sin 60 = 2/√3;
Sec 60 Degree = 1/Cos 60 = 2; and
Cot 60 Degree = 1/Tan 60 = 1/√3
Cracking the Code of Trigonometric Ratio: Special Angles 0° and 90°
The trigonometry ratio might sound intimidating, but there is no need to fear, 10th graders! This is because it's all about understanding the relationship between angles and sides in triangles to comprehend the trigonometric ratio/ trigonometric identities of 0 and 90 degrees for different angles.
With this in consideration, we'll tackle a special case: figuring out trigonometric ratios for angles of 0 degrees and 90 degrees.
Imagine that you have a right triangle like a slice of pizza (yum!) with a hypotenuse (the longest side), and the other two sides are opposite and adjacent to the right angle. Now, let us visualize this: what happens to these sides as we change the angle at the pointy corner?
In other words, using the sides and the angles, we would understand the connection between them in a right-angled triangle. The angles we will be using to decipher the connection between the angles and the sides are sine (sin), cosine (cos), tangent (tan), and more.
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Sin (this is the angle defined by opposite over hypotenuse): In this case, the opposite side becomes almost zero at a 0° angle, sin 0° is also practically zero.
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Cos (this is the angle defined by adjacent over hypotenuse): In this scenario, the adjacent side doesn’t change and remains the same length as the hypotenuse at 0°, so cos 0° is simply 1.
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Tan (this is the angle defined by sin over cos): We will get the value of Tan when we divide opposite over adjacent — hence, we divide sin by cos.
Remember, the value of sin is almost zero at 0°, and the value of cos is 1. Thus by dividing a tiny number by 1 gives us... almost zero and hence, tan 0° is also 0.
Similarly as done earlier, we can now find the values of Cosec, Sec, and Cot based on the values of Sin, Cos, and Cot as they have a relationship that has been explained before.
So, Cosec can be derived from the formula of 1/sin A, and in our case, it is 1/0 = Not Defined (or Unknown Value).
Sec A can be derived from the formula of 1/cos A and in our case, it is 1/1 = 1.
Cot A can be derived from the formula of 1/tan A and in our case, it is 1/0 = Again, Not Defined.