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Magazine Chapter 5: Problem 44
Prove each identity. $$\sin \left(90^{\circ}+x\right)-\sin \left(90^{\circ}-x\right)=0$$
Short Answer
Expert verified
The identity is true; both terms simplify to \(\cos x\), making their difference zero.
Step by step solution
01
Understand the Problem
We need to prove the trigonometric identity: \[\sin(90^\circ + x) - \sin(90^\circ - x) = 0.\]This means both expressions on the left-hand side must simplify to the same value, making their difference zero.
02
Apply Sine Angle Sum Identity
Using the sine angle sum identity: \[\sin(a + b) = \sin a \cos b + \cos a \sin b,\]we express \(\sin(90^\circ + x)\) as:\[\sin(90^\circ + x) = \sin 90^\circ \cos x + \cos 90^\circ \sin x.\] Since \(\sin 90^\circ = 1\) and \(\cos 90^\circ = 0\), we have: \[\sin(90^\circ + x) = 1 \cdot \cos x + 0 \cdot \sin x = \cos x.\]
03
Apply Sine Angle Difference Identity
Similarly, apply the sine angle difference identity: \[\sin(a - b) = \sin a \cos b - \cos a \sin b,\]to express \(\sin(90^\circ - x)\) as:\[\sin(90^\circ - x) = \sin 90^\circ \cos x - \cos 90^\circ \sin x.\]Using \(\sin 90^\circ = 1\) and \(\cos 90^\circ = 0\), it simplifies to:\[\sin(90^\circ - x) = 1 \cdot \cos x - 0 \cdot \sin x = \cos x.\]
04
Subtract and Simplify
Subtract \(\sin(90^\circ - x)\) from \(\sin(90^\circ + x)\):\[\cos x - \cos x = 0.\]This confirms the left-hand side of the identity equals zero.
05
Conclude the Proof
Since \(\sin(90^\circ + x) - \sin(90^\circ - x) = 0\), the identity is true. Both expressions on the left simplify to \(\cos x\), thus their difference is zero.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sine Angle Sum Identity
The sine angle sum identity is a fundamental concept in trigonometry. It allows us to break down a sine function of a sum of two angles, making complex calculations more manageable. The identity is expressed as follows: \[\sin(a + b) = \sin a \cos b + \cos a \sin b.\] This formula shows how the sine of the sum of two angles can be transformed into a sum of products of sines and cosines of the individual angles.
- Understanding Components: The identity involves sine and cosine functions of both angles \(a\) and \(b\). This way, both sine and cosine contribute to the expansion.
- Special Cases: This identity is versatile; for example, when either \(a\) or \(b\) is \(90^\circ\), some terms simplify due to the known values of \(\sin 90^\circ\) and \(\cos 90^\circ\).
Sine Angle Difference Identity
The sine angle difference identity is similar in structure to the angle sum identity, but focuses on the difference of two angles. It is expressed as:\[\sin(a - b) = \sin a \cos b - \cos a \sin b.\] This identity helps transform the sine of the difference between two angles into a subtraction of two products.
- Component Understanding: This identity also involves sine and cosine functions of both angles \(a\) and \(b\). The negative sign between the two products accounts for the difference in angles.
- Special Cases: As seen in the original problem with \(\sin(90^\circ - x)\), certain values simplify calculations greatly, highlighting the identity's utility for handling angles like \(90^\circ\).
Trigonometric Proofs
Trigonometric proofs involve demonstrating that an equation is true for all values within a certain domain. They are essential for validating trigonometric identities and concepts. In our exercise, we aimed to prove a trigonometric identity.Steps in Trigonometric Proofs:
- Understand the Identity: Before diving into calculations, comprehend what the identity is stating. This involves understanding the terms and their meanings.
- Apply Known Identities: Use substitutions from common identities (like sine angle sum and difference identities) to simplify expressions. These substitutions help break down complex trigonometric equations into simpler parts.
- Simplification: Subtract or add expressions carefully to verify the identity holds true. Simplifying often reveals that both sides of the equation are equivalent.
