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Without Using A Calculator Simplify Csc-1 Sec 5

Reviewed by Calculator Editorial Team

Simplifying trigonometric expressions like csc(1) sec(5) without a calculator requires understanding of fundamental trigonometric identities and properties. This guide will walk you through the process step by step, ensuring you can simplify such expressions accurately.

Introduction

The expression csc(1) sec(5) involves cosecant (csc) and secant (sec) functions evaluated at 1 and 5 radians, respectively. Simplifying this expression without a calculator requires applying trigonometric identities and understanding the relationships between these functions.

Cosecant is the reciprocal of sine, and secant is the reciprocal of cosine. Therefore, we can rewrite the expression in terms of sine and cosine:

csc(θ) = 1/sin(θ)

sec(θ) = 1/cos(θ)

Therefore, csc(1) sec(5) = (1/sin(1)) * (1/cos(5)) = 1/(sin(1) cos(5))

This form is already simplified, but we can explore further simplification using trigonometric identities.

Trigonometric Identities

Trigonometric identities can help simplify expressions involving sine and cosine functions. One useful identity is the product-to-sum formula:

sin(A) cos(B) = [sin(A+B) + sin(A-B)] / 2

Applying this to our expression:

sin(1) cos(5) = [sin(1+5) + sin(1-5)] / 2 = [sin(6) + sin(-4)] / 2

Since sin(-x) = -sin(x), this becomes:

[sin(6) - sin(4)] / 2

Therefore, the original expression can be rewritten as:

csc(1) sec(5) = 1 / [(sin(6) - sin(4)) / 2] = 2 / (sin(6) - sin(4))

Step-by-Step Solution

  1. Express csc(1) and sec(5) in terms of sine and cosine:

    csc(1) = 1/sin(1)

    sec(5) = 1/cos(5)

  2. Multiply the two expressions:

    csc(1) sec(5) = (1/sin(1)) * (1/cos(5)) = 1/(sin(1) cos(5))

  3. Apply the product-to-sum identity to sin(1) cos(5):

    sin(1) cos(5) = [sin(6) + sin(-4)] / 2 = [sin(6) - sin(4)] / 2

  4. Substitute back into the expression:

    csc(1) sec(5) = 1 / [(sin(6) - sin(4)) / 2] = 2 / (sin(6) - sin(4))

The simplified form of csc(1) sec(5) is 2 / (sin(6) - sin(4)).

Verification

To verify the simplification, let's compute the original expression and the simplified form numerically using approximate values:

sin(1) ≈ 0.8415

cos(5) ≈ 0.2837

Original expression: csc(1) sec(5) ≈ 1 / (0.8415 * 0.2837) ≈ 4.216

sin(6) ≈ -0.2794

sin(4) ≈ -0.7568

Simplified expression: 2 / (-0.2794 - (-0.7568)) ≈ 2 / (0.4774) ≈ 4.190

The results are close, with a slight difference due to rounding errors in the numerical approximations. This confirms that the simplification is correct.

Common Mistakes

When simplifying trigonometric expressions, several common mistakes can occur:

  • Incorrectly applying trigonometric identities, such as using the wrong product-to-sum formula.
  • Forgetting to consider the signs of trigonometric functions, especially when dealing with negative angles.
  • Miscounting the number of terms or signs when expanding expressions.
  • Overlooking the need to rationalize denominators or simplify fractions.

To avoid these mistakes, carefully review each step of the simplification process and verify the results numerically when possible.

FAQ

What is the simplified form of csc(1) sec(5)?
The simplified form is 2 / (sin(6) - sin(4)).
Can csc(1) sec(5) be simplified further?
No, the expression is already in its simplest form using basic trigonometric identities.
How do I verify the simplification is correct?
You can verify by computing both the original and simplified expressions numerically using approximate values for the trigonometric functions.
What identities are used to simplify csc(1) sec(5)?
The reciprocal identities (csc = 1/sin, sec = 1/cos) and the product-to-sum identity (sin(A)cos(B) = [sin(A+B) + sin(A-B)]/2) are used.
Are there any common mistakes to avoid when simplifying this expression?
Yes, common mistakes include incorrectly applying identities, forgetting sign considerations, and miscounting terms or signs.