Exact Value Of Sin(86)cos(26) - Cos(86)sin(26) Using Identities

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This article aims to provide a detailed explanation of how to find the exact value of the trigonometric expression: $ \operatorname{sin} 86^{\circ} \operatorname{cos} 26^{\circ} - \operatorname{cos} 86^{\circ} \operatorname{sin} 26^{\circ}

without using a calculator. We will leverage trigonometric identities, specifically the sine subtraction formula, to simplify the expression and arrive at an exact value. This method demonstrates a powerful technique in trigonometry for solving expressions that might seem complex at first glance. # Introduction In the realm of trigonometry, certain expressions can appear daunting and unsolvable without computational aids. However, a deep understanding of trigonometric identities allows us to simplify these expressions and find exact values. The expression at hand,

\operatorname{sin} 86^{\circ} \operatorname{cos} 26^{\circ} - \operatorname{cos} 86^{\circ} \operatorname{sin} 26^{\circ}

, is a classic example where the application of an appropriate identity can lead to a straightforward solution. This article will walk you through the step-by-step process of identifying the relevant identity, applying it to the expression, and arriving at the final answer. We will explore the sine subtraction formula, which is the key to unlocking this problem, and highlight its significance in simplifying trigonometric expressions. # Key Concepts: Trigonometric Identities Trigonometric identities are equations that are true for all values of the variables involved. These identities are essential tools for simplifying trigonometric expressions and solving trigonometric equations. Mastering these identities is crucial for anyone studying trigonometry and related fields. Some of the most commonly used identities include: * **Pythagorean Identities:** * $\operatorname{sin}^2(\theta) + \operatorname{cos}^2(\theta) = 1 

*   $1 + \operatorname{tan}^2(\theta) = \operatorname{sec}^2(\theta)$
*   $1 + \operatorname{cot}^2(\theta) = \operatorname{csc}^2(\theta)$
  • Angle Sum and Difference Identities:

    • sin⁡(A+B)=sin⁡(A)cos⁡(B)+cos⁡(A)sin⁡(B)\operatorname{sin}(A + B) = \operatorname{sin}(A)\operatorname{cos}(B) + \operatorname{cos}(A)\operatorname{sin}(B)

    • sin⁡(A−B)=sin⁡(A)cos⁡(B)−cos⁡(A)sin⁡(B)\operatorname{sin}(A - B) = \operatorname{sin}(A)\operatorname{cos}(B) - \operatorname{cos}(A)\operatorname{sin}(B)

    • cos⁡(A+B)=cos⁡(A)cos⁡(B)−sin⁡(A)sin⁡(B)\operatorname{cos}(A + B) = \operatorname{cos}(A)\operatorname{cos}(B) - \operatorname{sin}(A)\operatorname{sin}(B)

    • cos⁡(A−B)=cos⁡(A)cos⁡(B)+sin⁡(A)sin⁡(B)\operatorname{cos}(A - B) = \operatorname{cos}(A)\operatorname{cos}(B) + \operatorname{sin}(A)\operatorname{sin}(B)

  • Double Angle Identities:

    • sin⁡(2θ)=2sin⁡(θ)cos⁡(θ)\operatorname{sin}(2\theta) = 2\operatorname{sin}(\theta)\operatorname{cos}(\theta)

    • cos⁡(2θ)=cos⁡2(θ)−sin⁡2(θ)\operatorname{cos}(2\theta) = \operatorname{cos}^2(\theta) - \operatorname{sin}^2(\theta)

    • cos⁡(2θ)=2cos⁡2(θ)−1\operatorname{cos}(2\theta) = 2\operatorname{cos}^2(\theta) - 1

    • cos⁡(2θ)=1−2sin⁡2(θ)\operatorname{cos}(2\theta) = 1 - 2\operatorname{sin}^2(\theta)

In this particular problem, the sine subtraction identity is the most relevant. This identity states that:

sin⁡(A−B)=sin⁡(A)cos⁡(B)−cos⁡(A)sin⁡(B)\operatorname{sin}(A - B) = \operatorname{sin}(A)\operatorname{cos}(B) - \operatorname{cos}(A)\operatorname{sin}(B)

This identity allows us to combine two separate sine and cosine terms into a single sine term, which can greatly simplify expressions.

The given expression is:

sin⁡86∘cos⁡26∘−cos⁡86∘sin⁡26∘\operatorname{sin} 86^{\circ} \operatorname{cos} 26^{\circ} - \operatorname{cos} 86^{\circ} \operatorname{sin} 26^{\circ}

Comparing this expression with the sine subtraction identity:

sin⁡(A−B)=sin⁡(A)cos⁡(B)−cos⁡(A)sin⁡(B)\operatorname{sin}(A - B) = \operatorname{sin}(A)\operatorname{cos}(B) - \operatorname{cos}(A)\operatorname{sin}(B)

We can see a clear correspondence:

  • A=86∘A = 86^{\circ}

  • B=26∘B = 26^{\circ}

Thus, we can rewrite the expression using the identity as:

sin⁡(86∘−26∘)\operatorname{sin}(86^{\circ} - 26^{\circ})

Simplifying the angle inside the sine function:

sin⁡(60∘)\operatorname{sin}(60^{\circ})

The sine of $60^\circ}$ is a well-known value that can be derived from the properties of a 30-60-90 triangle. In such a triangle, the sides are in the ratio $1 \sqrt{3 : 2$, where the side opposite the 30-degree angle is 1, the side opposite the 60-degree angle is $\sqrt{3}$, and the hypotenuse is 2.

The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Therefore:

sin⁡(60∘)=oppositehypotenuse=32\operatorname{sin}(60^{\circ}) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}

Thus, the exact value of $\operatorname{sin}(60^{\circ})$ is $\frac{\sqrt{3}}{2}$. This value is a fundamental trigonometric ratio and should be memorized for quick recall in various mathematical problems.

Therefore, the exact value of the given expression is:

sin⁡86∘cos⁡26∘−cos⁡86∘sin⁡26∘=sin⁡(86∘−26∘)=sin⁡(60∘)=32\operatorname{sin} 86^{\circ} \operatorname{cos} 26^{\circ} - \operatorname{cos} 86^{\circ} \operatorname{sin} 26^{\circ} = \operatorname{sin}(86^{\circ} - 26^{\circ}) = \operatorname{sin}(60^{\circ}) = \frac{\sqrt{3}}{2}

This result is obtained without the use of a calculator, relying solely on trigonometric identities and the knowledge of standard trigonometric values.

In conclusion, by recognizing the structure of the given expression and applying the appropriate trigonometric identity, we successfully found the exact value of $ \operatorname{sin} 86^{\circ} \operatorname{cos} 26^{\circ} - \operatorname{cos} 86^{\circ} \operatorname{sin} 26^{\circ}

withoutresortingtoacalculator.Thekeytothissolutionwastheapplicationofthesinesubtractionidentity,whichallowedustosimplifytheexpressionto without resorting to a calculator. The key to this solution was the application of the sine subtraction identity, which allowed us to simplify the expression to

\operatorname{sin}(60^{\circ})

.Knowingtheexactvalueof. Knowing the exact value of

\operatorname{sin}(60^{\circ})

,whichis, which is

\frac{\sqrt{3}}{2}

, enabled us to arrive at the final answer. This exercise underscores the importance of mastering trigonometric identities. These identities are powerful tools that allow us to manipulate and simplify complex expressions, making them more manageable. The ability to recognize patterns and apply appropriate identities is a fundamental skill in trigonometry and is essential for solving a wide range of problems. By understanding and utilizing these identities, we can often find elegant solutions to problems that might otherwise seem intractable. This not only enhances our mathematical abilities but also deepens our appreciation for the beauty and coherence of trigonometric principles. # Keywords Trigonometric identities, sine subtraction formula, exact value, trigonometry,

\operatorname{sin}(A - B)

,,

\operatorname{sin} 60^{\circ}

,simplifyexpressions,anglesubtraction,trigonometricratios,mathematicalproblemsolving., simplify expressions, angle subtraction, trigonometric ratios, mathematical problem solving.