01 · Explore
The Concept of Trigonometry
Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of a triangle.
The word 'trigonometry' is derived from the Greek words 'tri' (meaning three), 'gon' (meaning sides), and 'metron' (meaning measure). Historically, it was used by early astronomers to find the distances of stars and planets from the Earth. Today, it remains essential in engineering and physical sciences.
In many real-life situations, such as finding the height of a tower like the Qutub Minar or the width of a river, we can imagine right-angled triangles. By using trigonometric techniques, we can calculate these heights and distances without physical measurement.
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What does the word 'trigonometry' literally mean based on its Greek roots?
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02 · Explore
Trigonometric Ratios of an Acute Angle
Trigonometric ratios express the relationship between an acute angle of a right triangle and the lengths of its sides.
Consider a right triangle ABC, right-angled at B. For the acute angle A, the side BC is the 'side opposite to angle A', AB is the 'side adjacent to angle A', and AC is the 'hypotenuse'.
The six trigonometric ratios for angle A are defined as: sine (sin A = opposite/hypotenuse), cosine (cos A = adjacent/hypotenuse), tangent (tan A = opposite/adjacent), cosecant (cosec A = 1/sin A), secant (sec A = 1/cos A), and cotangent (cot A = 1/tan A).
It is important to note that 'sin A' is a single symbol representing the sine of angle A; it is not the product of 'sin' and 'A'. Also, the values of these ratios depend only on the measure of the angle, not on the size of the triangle.
At boundary angles, use sine/cosine ratios with their domains. For instance, cot 90° = cos 90°/sin 90° = 0, even though tan 90° is undefined. You cannot use 1/tan A where tan A itself is undefined.
| Ratio | Abbreviation | Side Relationship |
|---|---|---|
| sine of A | sin A | BC/AC |
| cosine of A | cos A | AB/AC |
| tangent of A | tan A | BC/AB |
| cosecant of A | cosec A | AC/BC |
| secant of A | sec A | AC/AB |
| cotangent of A | cot A | AB/BC |
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If tan A = sin A / cos A, what is the ratio for cot A in terms of sin and cos?
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03 · Explore
Calculating Ratios from a Given Value
If one trigonometric ratio is known, the others can be determined using the Pythagoras Theorem.
When we are given a ratio like tan A = 4/3, we can treat the sides as 4k and 3k, where k is a positive constant. By applying the Pythagoras Theorem (AC² = AB² + BC²), we find the third side.
Once all three sides (opposite, adjacent, and hypotenuse) are expressed in terms of k, the k values cancel out when calculating any other trigonometric ratio.
Finding Ratios from tan A
tan A = 4/3; BC = 4k, AB = 3k. AC = √((4k)² + (3k)²) = √(16k² + 9k²) = √(25k²) = 5k.
Using the sides BC=4k, AB=3k, and AC=5k, we find sin A = BC/AC = 4k/5k = 4/5 and cos A = AB/AC = 3k/5k = 3/5.
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For an acute angle A, if sin A = 3/5, what is cos A?
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04 · Explore
Trigonometric Ratios of 45°
The ratios for 45° are derived from an isosceles right-angled triangle.
In a right triangle ABC where angle A = 45°, the third angle C must also be 45°. This makes the triangle isosceles, meaning AB = BC.
If we let AB = BC = a, then by Pythagoras Theorem, the hypotenuse AC = √(a² + a²) = a√2. Using these side lengths, we can define the exact values for all 45° ratios.
Values for 45°
sin 45° = a / (a√2) = 1/√2; cos 45° = a / (a√2) = 1/√2; tan 45° = a / a = 1.
Since the opposite and adjacent sides are equal, sine and cosine are identical, and tangent is exactly 1.
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What is the value of sec 45°?
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05 · Explore
Trigonometric Ratios of 30° and 60°
These ratios are derived by bisecting an equilateral triangle.
Consider an equilateral triangle ABC with side length 2a. Each angle is 60°. If we draw a perpendicular AD from A to BC, it bisects angle A into two 30° angles and bisects side BC into two segments of length 'a'.
In the resulting right triangle ABD, the hypotenuse AB = 2a and the base BD = a. Using Pythagoras Theorem, the height AD = √((2a)² - a²) = √(3a²) = a√3. We use these lengths to find the ratios for both 30° and 60°.
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 60° | √3/2 | 1/2 | √3 |
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Why is sin 30° equal to cos 60°?
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