Further Mathematics

Trigonometry

# Understanding Sine, Cosine, and Tangent Ratios

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Understanding Sine, Cosine, and Tangent Ratios

**Understanding Sine, Cosine, and Tangent Ratios**

**Introduction to Ratios in Trigonometry**

- Trigonometry is a branch of mathematics that studies triangles, specifically the relationships among their angles and sides.
- The three primary ratios in trigonometry are sine (sin), cosine (cos), and tangent (tan).
- These ratios are built upon using the right-angled triangle, where one angle is exactly 90 degrees.

**Sine Ratio**

- The sine of an angle is described as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
- Mathematical representation:
**sin(θ) = Opposite/Hypotenuse**. - Remember, the hypotenuse is the longest side of the right-angled triangle, always opposite the right angle.

**Cosine Ratio**

- The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.
- Mathematical representation:
**cos(θ) = Adjacent/Hypotenuse**. - Relevant to remember: the adjacent side is the side that forms the angle θ along with the hypotenuse.

**Tangent Ratio**

- The tangent of an angle is the ratio of the sine of the angle to the cosine of the angle. For a right-angled triangle, it's the ratio of the length of the opposite side to the length of the adjacent side.
- Mathematical representation:
**tan(θ) = sin(θ)/cos(θ)**or**tan(θ) = Opposite/Adjacent**. - Note this importance: the tangent ratio only makes sense as long as the cosine of the angle isn't zero to avoid division by zero.

**Usage and Practical Applications**

- These ratios are essential in countless situations where angles and distances need to be determined, such as in navigation, architecture, physics, engineering, and computer graphics.
- Calculators often have specific buttons for these trigonometric functions, enabling the computation of the ratios for any given angle.
- Sine and cosine ratios are crucial in a periodic phenomenon like sound and light waves, while tangent ratios predominantly find their use in slope calculations.

**Remember SOHCAHTOA**

- An easy way to remember the three basic functions sin, cos, and tan is by the mnemonic
**SOHCAHTOA**.- 'S' stands for Sine: Opposite/Hypotenuse.
- 'C' stands for Cosine: Adjacent/Hypotenuse.
- 'T' stands for Tangent: Opposite/Adjacent.

- This acronym is typically one of the most beneficial memorisation aids when first learning about trigonometric ratios.