Logarithm - 5 Combining Logarithm and Trigonometry

  • In previous lessons, we learned about logarithms and trigonometry separately.
  • In this lesson, we will explore how to combine logarithm and trigonometric functions.
  • We will understand how to solve equations involving logarithmic and trigonometric terms.
  • This will help us in solving more complex mathematical problems.
  • Let’s begin by revisiting the basic properties of logarithms and trigonometry.

Recap of Logarithmic Properties

  • Logarithm is the inverse function of exponentiation.
  • We use logarithms to solve exponential equations.
  • Here are some key properties of logarithms:
  1. Property of Logarithm of Product: log(ab) = log(a) + log(b)
  1. Property of Logarithm of Quotient: log(a/b) = log(a) - log(b)
  1. Property of Logarithm of Power: log(a^b) = b * log(a)
  1. Property of Logarithm of Base: log(a) = log(b) / log(c)
  • These properties will come in handy when combining logarithms with trigonometry.

Recap of Trigonometric Functions

  • Trigonometry deals with the relationships between the angles and sides of triangles.
  • The three primary trigonometric functions are:
  1. Sine (sin): sin(x) = opposite / hypotenuse
  1. Cosine (cos): cos(x) = adjacent / hypotenuse
  1. Tangent (tan): tan(x) = opposite / adjacent
  • These functions can be used to calculate angles and sides of triangles in various trigonometric equations.

Example 1: Combining Logarithm and Trigonometry

  • Let’s solve an equation that involves both logarithmic and trigonometric terms: log(sin(x)) = log(cos(x))
  • To solve this equation, we can use the property of logarithm of base.
  • Taking the logarithm of both sides, we get: log(sin(x)) / log(10) = log(cos(x)) / log(10)
  • Simplifying further, we obtain: log(sin(x)) = log(cos(x))
  • Now, we can solve for x using trigonometric identities.

Example 2: Combining Logarithm and Trigonometry

  • Let’s solve another equation involving logarithmic and trigonometric terms: log(sin(x)) + log(cos(x)) = 2
  • To solve this equation, we can use the property of logarithm of product.
  • Combining the logarithmic terms, we have: log(sin(x) * cos(x)) = 2
  • Taking the antilog (exponentiation) of both sides: sin(x) * cos(x) = 10^2
  • Simplifying further: sin(x) * cos(x) = 100
  • Now, we can solve for x using trigonometric identities.

Combining Logarithm and Trigonometry: Important Points to Remember

  • When combining logarithm and trigonometry, follow these key points:
  1. Apply logarithmic properties before or after solving trigonometric equations.
  1. Use trigonometric identities to simplify and solve equations involving logarithmic and trigonometric terms.
  1. Be cautious of domain restrictions and specific range values for the trigonometric functions.
  • Practice several examples to strengthen your understanding of combining logarithm and trigonometry.

Summary

  • In this lesson, we learned how to combine logarithm and trigonometry.
  • We recapped the properties of logarithms and the basic trigonometric functions.
  • We solved equations involving logarithmic and trigonometric terms using the properties and identities.
  • Remember to practice and familiarize yourself with various examples to improve problem-solving skills.

Slide 11

Combining Logarithm and Trigonometry

  • Property of Logarithm of Sum: log(a + b) ≠ log(a) + log(b)
  • Property of Logarithm of Difference: log(a - b) ≠ log(a) - log(b)
  • Property of Logarithm of Product: log(a * b) = log(a) + log(b)
  • Property of Logarithm of Quotient: log(a / b) = log(a) - log(b)
  • Property of Logarithm of Power: log(a^b) = b * log(a)

Slide 12

Combining Logarithm and Trigonometry

  • The inverse functions of the trigonometric functions are the arc trigonometric functions.
  • For example, the arcsine function is denoted as asin(x).
  • We can combine logarithmic and arc trigonometric functions to solve equations.

Slide 13

Example 1: Combining Logarithm and Trigonometry

  • Solve the equation log(sin(x)) = log(cos(x)).
  • Taking the logarithm on both sides gives: log(sin(x)) / log(10) = log(cos(x)) / log(10).
  • Simplifying further, we obtain: log(sin(x)) = log(cos(x)).
  • Now, we can solve for x using trigonometric identities.

Slide 14

Example 2: Combining Logarithm and Trigonometry

  • Solve the equation log(sin(x)) + log(cos(x)) = 2.
  • Combining the logarithmic terms gives: log(sin(x) * cos(x)) = 2.
  • Taking the antilog (exponentiation) of both sides gives: sin(x) * cos(x) = 10^2.
  • Simplifying further, we obtain: sin(x) * cos(x) = 100.
  • Now, we can solve for x using trigonometric identities.

Slide 15

Combining Logarithm and Trigonometry: Important Points

  • Apply logarithmic properties before or after solving trigonometric equations.
  • Use trigonometric identities to simplify and solve equations involving logarithmic and trigonometric terms.
  • Be cautious of domain restrictions and specific range values for the trigonometric functions.

Slide 16

Example 3: Combining Logarithm and Trigonometry

  • Solve the equation log(cos(x)) - log(tan(x)) = log(sin(x)).
  • Using the properties of logarithms, we can rewrite the equation as: log(cos(x) / tan(x)) = log(sin(x)).
  • Taking the antilog of both sides gives: cos(x) / tan(x) = sin(x).
  • Simplifying further, we obtain: cos(x) = sin(x) * tan(x).
  • Now, we can solve for x using trigonometric identities.

Slide 17

Example 4: Combining Logarithm and Trigonometry

  • Solve the equation log(tan(x)) - log(cot(x)) = log(sec(x)).
  • Using the properties of logarithms, we can rewrite the equation as: log(tan(x) / cot(x)) = log(sec(x)).
  • Taking the antilog of both sides gives: tan(x) / cot(x) = sec(x).
  • Simplifying further, we obtain: tan(x) = sec(x) * cot(x).
  • Now, we can solve for x using trigonometric identities.

Slide 18

Example 5: Combining Logarithm and Trigonometry

  • Solve the equation log(sec(x)) + log(cosec(x)) = log(tan(x)).
  • Combining the logarithmic terms gives: log(sec(x) * cosec(x)) = log(tan(x)).
  • Taking the antilog of both sides gives: sec(x) * cosec(x) = tan(x).
  • Simplifying further, we obtain: sec(x) = tan(x) / cosec(x).
  • Now, we can solve for x using trigonometric identities.

Slide 19

Practice Problems

  1. Solve the equation log(sin(x)) = log(cosec(x)).
  1. Solve the equation log(2sin(x) + 1) = log(sin(x)).
  1. Solve the equation log(cos(2x)) = log(1 - sin(x)). Remember to apply logarithmic and trigonometric properties as necessary.

Slide 20

Summary

  • Combined logarithm and trigonometry to solve equations.
  • Recalled properties of logarithms and arc trigonometric functions.
  • Solved examples involving logarithmic and trigonometric terms.
  • Emphasized importance of practicing various examples to enhance problem-solving skills.

Slide 21

Example 6: Combining Logarithm and Trigonometry

  • Solve the equation log(sin(2x)) = log(cos(2x)).
  • Taking the logarithm on both sides gives: log(sin(2x)) / log(10) = log(cos(2x)) / log(10).
  • Simplifying further, we obtain: log(sin(2x)) = log(cos(2x)).
  • Now, we can solve for x using trigonometric identities.

Slide 22

Example 7: Combining Logarithm and Trigonometry

  • Solve the equation log(sin^2(x)) = log(cos^2(x)).
  • Taking the logarithm on both sides gives: log(sin^2(x)) / log(10) = log(cos^2(x)) / log(10).
  • Simplifying further, we obtain: log(sin^2(x)) = log(cos^2(x)).
  • Now, we can solve for x using trigonometric identities.

Slide 23

Example 8: Combining Logarithm and Trigonometry

  • Solve the equation log(tan^3(x)) = log(sec^3(x)).
  • Taking the logarithm on both sides gives: log(tan^3(x)) / log(10) = log(sec^3(x)) / log(10).
  • Simplifying further, we obtain: log(tan^3(x)) = log(sec^3(x)).
  • Now, we can solve for x using trigonometric identities.

Slide 24

Domain Restrictions in Combining Logarithm and Trigonometry

  • When combining logarithm and trigonometry, it is important to consider the domain restrictions.
  • For example, the domain of the logarithmic function is restricted to positive numbers, i.e., log(x) is defined when x > 0.
  • Similarly, trigonometric functions have specific domain restrictions based on their properties.
  • It is crucial to take these domain restrictions into account while solving equations.

Slide 25

Range Values in Combining Logarithm and Trigonometry

  • The range values of trigonometric functions can affect the solutions of equations.
  • For example, the range of the cosine function is [-1, 1].
  • In equations involving logarithm and trigonometry, we need to ensure that the range values are consistent to obtain valid solutions.
  • Always consider the range values to ensure accurate solutions.

Slide 26

Important Properties: Logarithms

  • Property of Logarithm of Sum: log(a + b) ≠ log(a) + log(b)
  • Property of Logarithm of Difference: log(a - b) ≠ log(a) - log(b)
  • Property of Logarithm of Product: log(a * b) = log(a) + log(b)
  • Property of Logarithm of Quotient: log(a / b) = log(a) - log(b)
  • Property of Logarithm of Power: log(a^b) = b * log(a)

Slide 27

Important Properties: Trigonometry

  • Sine (sin): sin(x) = opposite / hypotenuse
  • Cosine (cos): cos(x) = adjacent / hypotenuse
  • Tangent (tan): tan(x) = opposite / adjacent
  • Arcsine (asin): asin(x) = angle whose sine is x
  • Arccosine (acos): acos(x) = angle whose cosine is x
  • Arctangent (atan): atan(x) = angle whose tangent is x

Slide 28

Key Points to Remember

  • Combine logarithm and trigonometry by utilizing the properties and identities of both functions.
  • Logarithmic properties can be applied before or after solving trigonometric equations.
  • Trigonometric identities help to simplify and solve equations involving logarithmic and trigonometric terms.
  • Consider domain restrictions and range values for accurate solutions.
  • Regular practice and understanding of properties and identities are crucial for success.

Slide 29

Conclusion

  • Combining logarithm and trigonometry expands the range of problems we can solve.
  • Understanding the properties and identities of both functions is essential.
  • Practice various examples to strengthen your problem-solving skills.
  • Remember the domain restrictions and range values while solving equations.
  • Continue exploring and studying mathematics to broaden your knowledge.

Slide 30

Questions?

  • Any questions or doubts regarding combining logarithm and trigonometry?
  • Feel free to ask for clarifications or additional examples.
  • Understanding this concept is crucial for your 12th Boards exam.
  • Practice regularly and seek help whenever needed.
  • Thank you for your attention!