IDENTITAS TRIGONOMETRI

Summary

This video provides an in-depth explanation of trigonometric identities and formulas essential for high school mathematics. It covers basic trigonometric ratios (sine, cosine, tangent, etc.), key identities like the Pythagorean identity, and their applications in solving trigonometric equations. The video includes clear, step-by-step examples of how to prove trigonometric expressions, reinforcing key concepts and encouraging viewers to practice problem-solving. By the end of the lesson, viewers gain a solid understanding of fundamental trigonometric relationships and their use in simplifying mathematical proofs.

Takeaways

• 😀 Understanding basic trigonometric ratios is essential for solving trigonometric problems (sin, cos, tan, sec, cosec, cot).
• 😀 Trigonometric functions can be derived using a right-angled triangle with Cartesian coordinates, where the hypotenuse is denoted by 'r'.
• 😀 The basic trigonometric formulas to memorize are: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent, etc.
• 😀 For easier memorization, associate each trigonometric function with a simple acronym or representation of its sides in a triangle.
• 😀 Reciprocal identities exist for the trigonometric functions: Cosec = 1/Sin, Sec = 1/Cos, Cot = 1/Tan.
• 😀 The first trigonometric identity is Pythagorean identity: Sin²a + Cos²a = 1.
• 😀 Proof of Pythagorean identity involves dividing the Pythagorean theorem (x² + y² = r²) by r² to form the identity.
• 😀 The second Pythagorean identity is: 1 + Tan²a = Sec²a, which can also be derived from the Pythagorean theorem.
• 😀 The third Pythagorean identity is: 1 + Cot²a = Cosec²a, derived similarly using the Pythagorean theorem.
• 😀 Various examples are provided to demonstrate how to simplify and prove trigonometric equations, such as simplifying expressions like Sin x Cos x / Cot x = Sin²x.

Q & A

What is the basic trigonometric identity for sine (sin)?

The basic trigonometric identity for sine (sin) is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle, or mathematically, sin(a) = y / r, where y is the length of the opposite side, and r is the hypotenuse.

How is the cosine (cos) identity derived?

The cosine (cos) identity is derived from the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It is mathematically expressed as cos(a) = x / r, where x is the length of the adjacent side, and r is the hypotenuse.

What is the relationship between sin(a) and cosec(a)?

Cosecant (cosec) is the reciprocal of sine. Therefore, cosec(a) = 1 / sin(a). It means that cosec(a) is the ratio of the hypotenuse to the opposite side of the right-angled triangle.

How is the secant (sec) identity related to cosine (cos)?

Secant (sec) is the reciprocal of cosine. Hence, sec(a) = 1 / cos(a), meaning secant represents the ratio of the hypotenuse to the adjacent side in a right-angled triangle.

What is the relationship between tangent (tan) and cotangent (cot)?

Tangent (tan) is the ratio of sine to cosine, or tan(a) = sin(a) / cos(a). Cotangent (cot) is the reciprocal of tangent, so cot(a) = 1 / tan(a), or equivalently cot(a) = cos(a) / sin(a).

Can you explain the Pythagorean identity involving sin²(a) and cos²(a)?

The Pythagorean identity states that sin²(a) + cos²(a) = 1. This identity holds true for any angle a, and it is derived from the Pythagorean theorem applied to a right-angled triangle.

What is the trigonometric identity involving tan²(a) and sec²(a)?

The trigonometric identity tan²(a) + 1 = sec²(a). This can be derived from the Pythagorean identity by dividing both sides by cos²(a), leading to tan²(a) = sin²(a) / cos²(a) and sec²(a) = 1 / cos²(a).

What is the identity that connects cot²(a) and cosec²(a)?

The identity cot²(a) + 1 = cosec²(a) is another Pythagorean identity. It can be derived similarly by dividing the Pythagorean identity by sin²(a).

How do you prove the trigonometric identity sin(x) * cos(x) / cot(x) = sin²(x)?

To prove sin(x) * cos(x) / cot(x) = sin²(x), we substitute cot(x) with cos(x) / sin(x). This gives us sin(x) * cos(x) / (cos(x) / sin(x)) = sin²(x), which simplifies to sin²(x), thus proving the identity.

How can the trigonometric expression √(1 - cos²(x)) / √(1 - sin²(x)) be simplified to tan(x)?

Using the identity sin²(x) + cos²(x) = 1, we can rewrite √(1 - cos²(x)) as sin(x) and √(1 - sin²(x)) as cos(x). Therefore, √(1 - cos²(x)) / √(1 - sin²(x)) = sin(x) / cos(x), which is equal to tan(x).