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Have you ever tried to find the height of a lighthouse without climbing it? Trigonometry makes it possible. By studying the relation between angles and sides of triangles, we can solve real-life and mathematical problems.
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The word trigonometry is derived from the Greek words 'trigonon,' meaning triangle, and 'metron,' meaning measure. The principles of trigonometry include the measurement of angles and problems involving angles. Three basic functions of trigonometry are sine, cosine, and tangent. These functions can be used to build other critical trigonometric functions such as cotangent, secant, and cosecant. It is important to know the three sides of a right-angled triangle to understand trigonometry. The three sides are:
The trigonometric functions are determined by using specific formulas. They are as follows:
\(Sin A = P / H \)
\(Cos A = B / H\)
\(Tan A = P/ B\)
\(Cosec A = H / P\)
\(Sec A = H / B\)
\(Cot A = B / P\)
Here, P, B, and H are perpendicular, base, and height of the right-angled triangle respectively.
The ancient Egyptians and Babylonians first developed trigonometry to help with practical tasks like building structures and studying the stars. However, Greek mathematician Hipparchus developed it as a mathematical discipline in the 2nd century BCE, creating the first trigonometric tables using chords.
The concepts of trigonometry refer to the important ideas that the subject covers. Let’s take a look at these concepts one by one.
Let's now understand the three sides of a right-angled triangle.
The ratios of sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle.
Let's meet the three most important ratios:
1. Sine (sin) - The height finder
Sine of an angle is defined by the ratio of length of sides which is opposite to the angle and the hypotenuse.
\( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
2. Cosine (cos) - The adjacent to the hypotenuse.
The cosine of an angle is defined by the ratio of lengths of sides which is adjacent to the angle and the hypotenuse.
\( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
3. Tangent (tan) - The angle decider
Tangent of an angle is defined by the ratio of the length of sides which is opposite to the angle and the side which is adjacent to the angle.
tan\(\theta\) = \(opposite\over adjacent\)
tan\(\theta\) = \(sin\over cos\)
Trigonometric functions are six basic functions where the input is the angle of a right triangle, and the output is a numerical value.
1. Cosecant (Cosec) - The reverse of sine
It is the reciprocal of sine, we flip the sine fraction. It is how we compare the hypotenuse to the opposite side.
\( cosec(\theta) = \frac{\text{Hypotenuse}}{\text{Opposite}} \;.\; cosec(\theta) = \frac{1}{\sin(\theta)} \)
2. Secant (sec) - The reverse of cosine
Secant is the reciprocal of cosine, we flip the cosine fraction. It is how we compare the hypotenuse to the adjacent side.
\( \sec(\theta) = \frac{\text{Hypotenuse}}{\text{Adjacent}} \;.\; \sec(\theta) = \frac{1}{\cos(\theta)} \)
3. Cotangent (cot) - The reverse of tangent
Cotangent is the reciprocal of tangent. It is how we compare the adjacent side to the opposite side.
\( \cot(\theta) = \frac{\text{Adjacent}}{\text{Opposite}} \;.\; \cot(\theta) = \frac{1}{\tan(\theta)} \)
Odd functions: In trigonometry, we call a function f(x) odd if the function satisfies the following property: f(−x) = −f(x). This means that flipping the input will also cause a flip in the output.
Even functions: We call a function f(x) even if it satisfies the property f(−x) = f(x). Here, flipping the input doesn’t change the output.
Trigonometric ratios can be applied to common angles. The table given below shows the value of trigonometric ratios at certain angles.
Let's learn about how to create a trigonometric ratio table.
Step 1: Write the first five whole numbers
Let’s start by writing the first five whole numbers. Imagine we are counting in a game of hopscotch, with some spaces in between each number so we can move to the next position.
0 | 1 | 2 | 3 | 4
Step 2: Divide each number by 4
Now, we divide each number by 4. Here, the number is divided by 4 to make it easier for students to understand.
\( 0 \;\big|\; \tfrac{1}{4} \;\big|\; \tfrac{2}{4} \;\big|\; \tfrac{3}{4} \;\big|\; \tfrac{4}{4} \)
When we simplify \({2 \over 4}\) and \({4 \over 4}\), we write it as:
\( 0 \;\big|\; \tfrac{1}{4} \;\big|\; \tfrac{1}{2} \;\big|\; \tfrac{3}{4} \;\big|\; 1 \)
Step 3: Take the square root of each resulting number
\( 0 \;\big|\; \sqrt{\tfrac{1}{4}} \;\big|\; \sqrt{\tfrac{1}{2}} \;\big|\; \sqrt{\tfrac{3}{4}} \;\big|\; 1 \)
Simplifying this we get,
Step 4: Sine values for angles 0°, 30°, 45°, 60°, and 90°
\( \sin(\theta) : \; 0 \;\big|\; \tfrac{1}{2} \;\big|\; \tfrac{1}{\sqrt{2}} \;\big|\; \tfrac{\sqrt{3}}{2} \;\big|\; 1 \)
Step 5: Reverse the order for cosine values
Now, let's have some fun and reverse the order of sine values to find the cosine values.
\( \: \; 1 \;\big|\; \frac{\sqrt{3}}{2} \;\big|\; \frac{1}{\sqrt{2}} \;\big|\; \frac{1}{2} \;\big|\; 0 \)
Step 6: Find the Tangent values
Now we find the tangent values, which are simply the sine that is divided by the cosine.
Tan 0° = 0/1 = 0
\( \tan 30^\circ = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} \)
\( \tan 45^\circ = \frac{1/\sqrt{2}}{1/\sqrt{2}} = 1 \)
\( \tan 60^\circ = \frac{\sqrt{3}/2}{1/2} = \sqrt{3} \)
tan 90° = 1/0 = Not defined.
Trigonometry formulas are used to solve trigonometric problems. These problems may include trigonometric ratios like sin, cos, tan, sec, cosec, and cot. Memorizing these mathematic formulas in trigonometry will help students solve problems easily.
1. Pythagoras trigonometric identities
\(\sin^2 \theta + \cos^2 \theta = 1\\1 + \tan^2 \theta = \sec^2 \theta\\\csc^2 \theta = 1 + \cot^2 \theta \)
2. Double angle identities
\( \begin{aligned} \sin(2\theta) &= 2 \sin \theta \cos \theta \\ \cos(2\theta) &= \cos^2 \theta - \sin^2 \theta \\ \tan(2\theta) &= \frac{2 \tan \theta}{1 - \tan^2 \theta} \end{aligned} \)
3. Product - Sum identities
\( \begin{aligned} \sin A + \sin B &= 2 \, \sin\left(\frac{A+B}{2}\right) \, \cos\left(\frac{A-B}{2}\right) \\ \sin A - \sin B &= 2 \, \cos\left(\frac{A+B}{2}\right) \, \sin\left(\frac{A-B}{2}\right) \\ \cos A + \cos B &= 2 \, \cos\left(\frac{A+B}{2}\right) \, \cos\left(\frac{A-B}{2}\right) \\ \cos A - \cos B &= -2 \, \sin\left(\frac{A+B}{2}\right) \, \sin\left(\frac{A-B}{2}\right) \end{aligned} \)
4. Sum and difference identities
\( \begin{aligned} \sin(A+B) &= \sin A \cos B + \cos A \sin B\\sin(A-B) &= \sin A \cos B - \cos A \sin B\\cos(A+B) &= \cos A \cos B - \sin A \sin B\\cos(A-B) &= \cos A \cos B + \sin A \sin B\\tan(A+B) &= \frac{\tan A + \tan B}{1 - \tan A \tan B}\\tan(A-B) &= \frac{\tan A - \tan B}{1 + \tan A \tan B} \end{aligned} \)
Trigonometry is a field of math that deals with the relationship between angles and sides of triangles. It has many practical applications, from measuring heights and distances to understanding sound and light waves. Trigonometry is applied in various fields.
Here are some fun and relatable real-life applications of trigonometry:
Architects use trigonometry to design buildings and structures. They calculate angles and lengths to ensure everything fits together properly.
Pilots and sailors rely on trigonometry for navigation.
Trigonometry is important for engineers to design machines and structures.
Trigonometric calculations are also used in astronomy and sports.
Trigonometry is all about angles and sides of a right-angled triangle. Kids can easily get confused by it. To master trigonometry, you can follow certain tips and tricks. In this section, let’s learn a few trigonometry tips and tricks.
1. SOH-CAH-TOA helps you remember the basics of sine, cosine, and tangent ratios:
SOH: sine = opposite/hypotenuse
CAH: cosine = adjacent/hypotenuse
TOA: tangent = opposite/adjacent
2. You can also create a fun sentence/phrase to memorize, like “Silly Owls Help Cats And Turtles”.
3. You can also make a trigonometric ratio triangle.
4. Draw a triangle and label its sides as opposite, adjacent, and hypotenuse.
5. Use color coding for different ratios and functions.
Hey kids! Trigonometry can be a little tricky sometimes, and we tend to make mistakes. Don't worry, it happens to everyone. Let's get to know some common mistakes that students often make in trigonometry and how to avoid them.
Find Sin 30° + cos 60°.
\(sin30^∘+cos60^∘=1\)
We know that,
sin 30° = 1/2, cos 60° = 1/2
Add them,
sin 30° + cos 60°
Simplify the values
1/2 + 1/2 = 1
\(sin30^∘+cos60^∘=1\)
If tan θ = 1, find θ in degrees.
θ = 45°
We know that,
tan θ = 1
So, tan \(45^∘\) = 1
Therefore,
θ = 45°
Prove that sin² 45° + cos² 45° = 1
\(sin^245^∘+cos^245^∘=1\)
We know that,
sin 45° = 1/2, cos 45° = 1/2
By using the Pythagorean identity.
sin2 45° + cos2 45°
Substituting the values,
= (1/√2)2 + (1/√2)2
= 1/2 + 1/2 = 1
\(sin^245^∘+cos^245^∘=1\)
If sin A = 5/13, find sec A.
\( \cos A = \frac{12}{13}, \quad \sec A = \frac{13}{12} \)
We know that,
sin2 A + cos2 A = 1
cos2A = 1 – (5/13)2
= 1 – (25/169)
Converting 1 to have the same denominator,
1 = 169/169
So the equation becomes,
= (169/169) - (25/169)
= 144/169
cos A = 12/13
\( \cos A = \frac{12}{13}, \quad \sec A = \frac{13}{12} \)
Prove that 1 + tan²θ = sec² θ
\(1+tan^2θ=sec^2θ\)
We know that,
tan θ = sin θ / cos θ
Square tan θ
\( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \)
Add 1
1 + tan2 θ = 1 + sin2 θ / cos2 θ
Write with a common denominator
\( 1 + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\cos^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta} \)
Now, use Pythagorean identity
\( \sin^2 \theta + \cos^2 \theta = 1 \)
\( \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} = \sec^2 \theta \)
Therefore, we get \(1+tan^2θ=sec^2θ\)
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Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
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