Evaluate Cot 40 * Tan 50 + 1/2 * Cos 35 / Sin 55

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Let's break down this trigonometric expression step by step. We aim to evaluate the expression:

cot 40 * tan 50 + 1/2 * cos 35 / sin 55

Understanding the Trigonometric Identities

To solve this, we'll use some fundamental trigonometric identities. The key ones are:

  1. Complementary Angle Identities: These identities relate trigonometric functions of complementary angles (angles that add up to 90 degrees). Specifically:

    • tan(90 - θ) = cot(θ)
    • cot(90 - θ) = tan(θ)
    • sin(90 - θ) = cos(θ)
    • cos(90 - θ) = sin(θ)

  2. Reciprocal Identities: These define relationships between trigonometric functions:

    • cot(θ) = 1 / tan(θ)

Step-by-Step Evaluation

Part 1: cot 40 * tan 50

First, notice that 40 and 50 are complementary angles, meaning 40 + 50 = 90. We can rewrite tan 50 using the complementary angle identity:

tan 50 = tan(90 - 40) = cot 40

Now, substitute this back into the first part of the expression:

cot 40 * tan 50 = cot 40 * cot 40

Part 2: 1/2 * cos 35 / sin 55

Similarly, 35 and 55 are complementary angles, since 35 + 55 = 90. We can rewrite sin 55 using the complementary angle identity:

sin 55 = sin(90 - 35) = cos 35

Substitute this back into the second part of the expression:

1/2 * cos 35 / sin 55 = 1/2 * cos 35 / cos 35

Since cos 35 / cos 35 = 1, this simplifies to:

1/2 * 1 = 1/2

Combining the Results

Now, let's put both parts back together:

cot 40 * tan 50 + 1/2 * cos 35 / sin 55 = cot 40 * cot 40 + 1/2

Rewriting cot 40 * tan 50 correctly

Oops! Let's correct our initial substitution. We correctly identified that tan 50 = cot 40. Thus:

cot 40 * tan 50 = cot 40 * cot 40 = cot^2 40 is incorrect.

Going back to the original expression:

cot 40 * tan 50

Using the complementary angle identity tan 50 = cot(90 - 50) = cot 40 is not correct. Instead:

tan 50 = tan(90 - 40) = cot 40

So,

cot 40 * tan 50 = cot 40 * cot 40 = 1 because tan 50 = cot (90-50) = cot 40, and cot x * tan x = 1

Substituting back

So the expression becomes:

1 + 1/2

Final Calculation

Now, we just need to add the two parts:

1 + 1/2 = 3/2

Therefore, the final answer is:

cot 40 * tan 50 + 1/2 * cos 35 / sin 55 = 3/2

Conclusion

By using trigonometric identities, particularly the complementary angle identities, we simplified the original expression step by step. Breaking down the problem into smaller parts made it easier to manage and solve. The final result is 3/2.

Keywords: Trigonometric identities, complementary angles, cotangent, tangent, sine, cosine, evaluating expressions.

Extra Exercises for Practice

Exercise 1: Simplify sin(x) * cos(90 - x) + cos(x) * sin(90 - x)

Solution:

Using complementary angle identities:

  • cos(90 - x) = sin(x)
  • sin(90 - x) = cos(x)

Substitute these into the expression:

sin(x) * sin(x) + cos(x) * cos(x) = sin^2(x) + cos^2(x)

By the Pythagorean identity, sin^2(x) + cos^2(x) = 1. Therefore, the simplified expression is 1.

Exercise 2: Evaluate tan(20) * tan(70)

Solution:

Recognize that 20 and 70 are complementary angles. Thus, tan(70) = tan(90 - 20) = cot(20). So the expression becomes:

tan(20) * cot(20)

Since cot(θ) = 1/tan(θ), we have tan(20) * (1/tan(20)) = 1. The result is 1.

Exercise 3: Simplify (cos(θ) / sin(θ)) / cot(θ)

Solution:

First, recognize that cos(θ) / sin(θ) = cot(θ). The expression then simplifies to:

cot(θ) / cot(θ) = 1

Thus, the simplified expression is 1.

Exercise 4: Evaluate (sin 45 / cos 45) + tan 45

Solution:

We know that sin 45 = cos 45, so sin 45 / cos 45 = 1. Also, tan 45 = 1. Therefore:

(sin 45 / cos 45) + tan 45 = 1 + 1 = 2

The result is 2.

Exercise 5: Simplify cos(60) * sin(30) + sin(60) * cos(30)

Solution:

We can use the sine addition formula: sin(A + B) = sin(A)cos(B) + cos(A)sin(B). In this case, A = 60 and B = 30. Thus:

cos(60) * sin(30) + sin(60) * cos(30) = sin(60 + 30) = sin(90) = 1

The result is 1.

Advanced Tips for Trigonometric Simplification

  1. Recognize Complementary Angles: Always look for angle pairs that add up to 90 degrees. This allows you to use complementary angle identities effectively.

  2. Use Pythagorean Identities: Identities like sin^2(x) + cos^2(x) = 1, 1 + tan^2(x) = sec^2(x), and 1 + cot^2(x) = csc^2(x) are powerful tools for simplifying expressions.

  3. Convert to Sine and Cosine: When in doubt, convert all trigonometric functions to sine and cosine. This often makes it easier to see how to simplify the expression.

  4. Apply Sum and Difference Formulas: Formulas like sin(A ± B) and cos(A ± B) can help simplify expressions involving sums or differences of angles.

  5. Practice Regularly: The more you practice, the quicker you'll become at recognizing patterns and applying the appropriate identities.

By mastering these techniques, you can tackle a wide range of trigonometric problems with confidence.

Keywords: Trigonometric simplification, advanced techniques, sine and cosine, Pythagorean identities, sum and difference formulas, complementary angles.

Remember, tackling these problems involves recognizing key relationships and applying trigonometric identities strategically. Keep practicing, and you'll become more comfortable and confident in your ability to solve these kinds of expressions!