RRB JE 2019 CLASSES/RRB NTPC 2019 /SSC CGL/SSC CHSL -MATHEMATICS CLASSES- 7-TRIGONOMETRY 3

Summary

In this trigonometry lecture, the instructor, Barat, explains various techniques for solving complex trigonometric problems. Topics include multiplication of tangent and cosine functions, using identities like sin² + cos² = 1, and solving infinite series problems. The lecture also covers applications of arithmetic and geometric progressions in trigonometry, along with their integration in solving problems. The importance of understanding logarithms, trigonometric identities, and geometric series for solving problems efficiently is emphasized, and various examples help reinforce these concepts. The session also highlights using multiple formulas for easy problem-solving and encourages practice for mastery.

Takeaways

• 😀 Trigonometric problems often require knowledge of identity pairs, such as using tan 90° - x = cot x and cot 90° - x = tan x.
• 😀 For the product of tan values, special trigonometric identities are used to simplify calculations, leading to answers like 1 or 0 in some cases.
• 😀 In series of trigonometric functions, multiplying terms can lead to simplifications when certain angles produce values like infinity or zero (e.g., tan 90° = ∞, cos 90° = 0).
• 😀 Arithmetic progressions are often applied to solve trigonometric series by identifying identity pairs (e.g., sin²θ + cos²θ = 1).
• 😀 The sum of terms in trigonometric series often involves breaking down angles into complementary pairs (e.g., sin 1° with sin 89°, tan 43° with tan 47°).
• 😀 Identity pairs, such as sin²θ + cos²θ = 1, help in simplifying trigonometric sums, allowing terms like sin²1° and sin²89° to cancel out.
• 😀 Cosine and sine values of complementary angles (e.g., cos(90° - x) = sin x) are key to simplifying trigonometric series.
• 😀 For infinite trigonometric series (e.g., sinθ + sin²θ + sin³θ), the sum can be calculated using the formula for geometric progressions, yielding results like sin 60° = √3/2.
• 😀 Logarithmic identities can be used to simplify sums of trigonometric functions (e.g., log(tan1°) + log(tan2°) + ... = log(1)) to find the result of such series.
• 😀 It's essential to understand the interplay between trigonometry and other mathematical concepts, like algebra, geometric progressions, and logarithms, to solve complex problems efficiently.

Q & A

What is the core topic of this video?

The core topic of the video is trigonometry, specifically focusing on various problems involving trigonometric functions like tangent, sine, cosine, and their relationships.

How does the speaker handle the multiplication of tangent functions involving unknown values?

The speaker uses the fact that certain angle pairs, such as 4° and 86°, or 43° and 47°, add up to 90°, which allows for simplification using the identity tan(90° - x) = cot(x). This leads to cancellation of terms and simplifies the calculation.

What happens when multiplying a series of cosines, from cos(1°) to cos(179°)?

The presence of cos(90°) in the multiplication causes the entire product to be zero because cos(90°) equals 0, and multiplying by 0 results in 0.

Why is the answer to the multiplication problem tan(10°) * tan(20°) * tan(30°) * ... tan(90°) zero?

The multiplication involves tan(90°), which is undefined or infinite. When an infinite value is multiplied by zero (from tan(90°)), the result is zero.

What is the significance of the identity sin²(θ) + cos²(θ) = 1 in solving trigonometric problems?

This identity helps in simplifying trigonometric expressions by converting sin²(θ) to cos²(θ) and vice versa, especially when working with sums of squares of sines and cosines in sequences.

How do you approach solving problems that involve a series of sine squares, such as sin²(1°) + sin²(2°) + ... + sin²(89°)?

The approach involves recognizing patterns or using trigonometric identities to pair terms such as sin²(θ) and cos²(90° - θ) to simplify the sum. Arithmetic progressions help in calculating the number of terms and pairs.

What is the importance of understanding arithmetic progressions in trigonometry?

Understanding arithmetic progressions is important because many trigonometric series problems, like summing sine and cosine squares, can be simplified using the formulas for the sum of terms in an arithmetic progression.

How is the value of sin²(45°) handled in the context of a sum of sine squares?

sin²(45°) is equal to 1/2, and it is the only term that does not pair with a cosine square term. It is added separately after calculating the sum of paired terms.

What formula is used to find the sum of an infinite geometric series in trigonometric problems?

The formula used is S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio. In the case of the infinite trigonometric series, the common ratio is sin(θ).

What is the result of the logarithmic trigonometric series log(tan(1°)) + log(tan(2°)) + ... + log(tan(89°))?

The logarithmic series simplifies to log(1), which equals zero because tan(1°) pairs with cot(1°), tan(2°) pairs with cot(88°), and so on, ultimately leading to the product being 1. The logarithm of 1 is 0.