**Title: Trigonometric Ratios (Sine, Cosine, Tangent)**
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Trigonometric ratios are fundamental functions used to relate the angles of a right-angled triangle to the lengths of its sides. The primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).
These ratios are essential in solving problems involving right-angle triangles, and they are widely applied in physics, engineering, astronomy, architecture, and design.
In a right triangle:
– Sine is defined as the ratio of the opposite side to the hypotenuse.
– Cosine is the ratio of the adjacent side to the hypotenuse.
– Tangent is the ratio of the opposite side to the adjacent side.
Formally, using angle \( \theta \), the definitions are:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}, \quad
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Let’s explore each of the trigonometric ratios with examples and solutions.
—
Sine (sin)
The sine ratio helps determine the height of an object using the angle and hypotenuse — very useful in vertical measurements.
**Easy Questions:**
1. If \( \theta = 30^\circ \), find \( \sin(\theta) \).
\[
\sin(30^\circ) = \frac{1}{2}
\]
2. In a right triangle, if the opposite side is 4 and the hypotenuse is 5, find \( \sin(\theta) \).
\[
\sin(\theta) = \frac{4}{5}
\]
**Medium Questions:**
1. A ladder is leaned against a wall making an angle of \( 45^\circ \) with the ground. Find the height it reaches if the ladder is 10 units long.
\[
\text{Height} = 10 \cdot \sin(45^\circ) = 10 \cdot \frac{\sqrt{2}}{2} = 5\sqrt{2}
\]
2. Given that \( \sin(\theta) = 0.6 \), find the angle \( \theta \) (in degrees).
\[
\theta = \sin^{-1}(0.6) \approx 36.87^\circ
\]
**Hard Questions:**
1. A tower casts a shadow of 15 meters when the angle of elevation of the sun is \( 53^\circ \). Find the height of the tower.
\[
\sin(53^\circ) = \frac{h}{15} \Rightarrow h = 15 \cdot \sin(53^\circ) \approx 15 \cdot 0.7986 \approx 11.979
\]
2. Verify identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \) when \( \theta = 60^\circ \)
\[
\sin^2(60^\circ) + \cos^2(60^\circ) = \left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = \frac{3}{4} + \frac{1}{4} = 1
\]
—
Cosine (cos)
Cosine is widely used in problems involving horizontal distances and in vector projections in physics and geometry.
**Easy Questions:**
1. If the adjacent side is 3 and the hypotenuse is 5, find \( \cos(\theta) \).
\[
\cos(\theta) = \frac{3}{5}
\]
2. Find \( \cos(60^\circ) \)
\[
\cos(60^\circ) = \frac{1}{2}
\]
**Medium Questions:**
1. A person sees a building at an angle of depression of \( 41^\circ \). If the observer is 50 m away horizontally, how far is the top of the building from the observer?
\[
\cos(41^\circ) = \frac{50}{h} \Rightarrow h = \frac{50}{\cos(41^\circ)} \approx \frac{50}{0.7547} \approx 66.27 \text{ m}
\]
2. Given \( \cos(\theta) = 0.8 \), find the angle \( \theta \).
\[
\theta = \cos^{-1}(0.8) \approx 36.87^\circ
\]
**Hard Questions:**
1. A diagonal of a rectangle makes an angle of \( 28^\circ \) with the base. The diagonal is 10 cm. Find the base of the rectangle.
\[
\cos(28^\circ) = \frac{\text{base}}{10} \Rightarrow \text{base} = 10 \cdot \cos(28^\circ) \approx 10 \cdot 0.8829 = 8.829 \text{ cm}
\]
2. In triangle ABC, angle C is \( 90^\circ \), side AB = 13, and AC = 5. Find \( \cos(B) \).
\[
\cos(B) = \frac{AC}{AB} = \frac{5}{13}
\]
—
Tangent (tan)
Tangent is incredibly useful in slope calculations, height and distance problems, and in navigation to determine heading angles.
**Easy Questions:**
1. If the opposite side is 4 and the adjacent is 3, find \( \tan(\theta) \).
\[
\tan(\theta) = \frac{4}{3}
\]
2. Evaluate \( \tan(45^\circ) \)
\[
\tan(45^\circ) = 1
\]
**Medium Questions:**
1. A cliff is observed from a distance of 80 meters. If the angle of elevation is \( 30^\circ \), find the height of the cliff.
\[
\tan(30^\circ) = \frac{h}{80} \Rightarrow h = 80 \cdot \tan(30^\circ) = 80 \cdot \frac{\sqrt{3}}{3} \approx 46.19
\]
2. Given \( \tan(\theta) = 2.5 \), find the angle \( \theta \).
\[
\theta = \tan^{-1}(2.5) \approx 68.2^\circ
\]
**Hard Questions:**
1. Verify the identity: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) for \( \theta = 30^\circ \)
\[
\tan(30^\circ) = \frac{\sin(30^\circ)}{\cos(30^\circ)} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}
\]
2. If a building 100 m tall casts a shadow of 50 m, find the angle of elevation to the top of the building.
\[
\tan(\theta) = \frac{100}{50} = 2 \Rightarrow \theta = \tan^{-1}(2) \approx 63.43^\circ
\]
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Importance of Studying Trigonometric Ratios
Studying trigonometric ratios equips learners with the ability to solve real-world problems in science, engineering, and technology — from designing mechanical machines to calculating distances in space.
Recent explorations by mathematicians such as Dr. Maryna Viazovska and Professor Terence Tao have ventured into high-dimensional trigonometric applications and advanced harmonic analysis, showing that fundamentals like sine, cosine, and tangent continue to evolve into complex mathematics.
Understanding trigonometric ratios forms the foundation for advanced topics such as Fourier series, waves, oscillations, and quantum mechanics.
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