A-Level Maths: E5-01 [Trigonometric Identities: Proving tanθ = sinθ / cosθ]

Summary

The video script introduces the fundamental concepts of trigonometry by using a right-angled triangle to define the angles and sides. It explains the three primary trigonometric ratios: sine, cosine, and tangent, in relation to the triangle's angles and sides. The sine of an angle (θ) is the ratio of the opposite side to the hypotenuse, while the cosine is the ratio of the adjacent side to the hypotenuse. The tangent of θ, which is the ratio of the opposite side to the adjacent side, is shown to be equivalent to the ratio of sine over cosine (tan(θ) = sin(θ)/cos(θ)). The script also highlights the importance of this identity in trigonometry and notes the special case where cosine of θ is zero, which leads to vertical asymptotes in the tangent function's graph. This summary provides an engaging overview of the key points discussed in the script, emphasizing the significance of these trigonometric identities in mathematical analysis.

Takeaways

• 📐 In a right-angled triangle, angle theta is used to define the sides as opposite, adjacent, and hypotenuse.
• 🧮 Sine (sin) of theta is the ratio of the opposite side to the hypotenuse.
• 📏 Cosine (cos) of theta is the ratio of the adjacent side to the hypotenuse.
• ↗️ Tangent (tan) of theta is the ratio of the opposite side to the adjacent side.
• 🔄 By dividing both the numerator and the denominator of tan theta by the hypotenuse, we get sin theta over cos theta.
• 🌟 The resulting expression tan theta = sin theta / cos theta is a fundamental trigonometric identity.
• ⚠️ The identity tan theta = sin theta / cos theta is undefined when cos theta equals zero.
• 🚫 When cos theta is zero, it results in vertical asymptotes for the tangent function graph.
• 📉 The vertical asymptotes occur where the cosine of theta is zero, which are important to recognize in trigonometric functions.
• 🔢 Understanding the relationship between sine, cosine, and tangent is crucial for further study in trigonometry.
• 🧹 The script emphasizes the need to learn and utilize these trigonometric identities for problem-solving in mathematics.
• 🔄 Dividing by the hypotenuse in the tangent ratio is a method to derive the trigonometric identity.

Q & A

What is the significance of the angle theta in the context of the triangle mentioned in the script?

Angle theta is the right angle in the triangle, which is used to define the relationships between the sides of the triangle in terms of trigonometric ratios.

What are the three sides of a right-angled triangle in relation to angle theta?

The three sides are the opposite side (opposite to angle theta), the adjacent side (next to angle theta), and the hypotenuse (the longest side opposite the right angle).

What does the sine function (sin) represent in trigonometry?

The sine function represents the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle.

What does the cosine function (cos) represent in trigonometry?

The cosine function represents the ratio of the length of the side adjacent to the angle to the length of the hypotenuse in a right-angled triangle.

What does the tangent function (tan) represent in trigonometry?

The tangent function represents the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a right-angled triangle.

What is the trigonometric identity that relates sine and cosine to tangent?

The trigonometric identity is tan(theta) = sin(theta) / cos(theta), which shows the relationship between the tangent of an angle and the sine and cosine of the same angle.

Why is it not possible to define tan(theta) when cos(theta) is zero?

When cos(theta) is zero, it implies that the side adjacent to the angle is of zero length, which is not possible in a triangle, hence tan(theta) is undefined as division by zero is not allowed in mathematics.

What are vertical asymptotes in the context of the tangent function?

Vertical asymptotes are the vertical lines on the graph of the tangent function where the function becomes undefined because cos(theta) is zero, causing the tangent to approach infinity or negative infinity.

Why is it important to learn and understand trigonometric identities?

Trigonometric identities are crucial for solving various mathematical problems involving triangles and angles, and they are fundamental in calculus, physics, engineering, and other scientific disciplines.

How can the understanding of trigonometric ratios help in real-world applications?

Trigonometric ratios are used in a wide range of applications, including navigation, physics for calculating forces, engineering for designing structures, and even in computer graphics for modeling 3D shapes.

What are the potential issues that can arise if we attempt to divide both the numerator and denominator of a fraction by zero?

Dividing by zero is undefined in mathematics because it leads to an indeterminate form and can result in division errors or paradoxes. It is a fundamental rule that you cannot divide by zero.

What is the role of the hypotenuse in the context of the trigonometric ratios discussed in the script?

The hypotenuse is the longest side of a right-angled triangle and serves as the reference side in defining the sine, cosine, and tangent ratios. It is the denominator in both sine and cosine functions and is used to normalize the ratios to a common scale.

How does the concept of a right-angled triangle provide a foundation for understanding trigonometry?

A right-angled triangle provides a geometric context for understanding the relationships between the angles and the sides of a triangle. These relationships, expressed as trigonometric ratios, form the basis for trigonometry and are essential for understanding more complex mathematical and real-world problems.