Prove that: [tex]\frac{\sin 20^{\circ}-\sin 10^{\circ}}{1-\cos 10^{\circ}+\cos 20^{\circ}}=\tan 10^{\circ}[/tex]
Answer (2)
The trigonometric identity 1 − c o s 1 0 ∘ + c o s 2 0 ∘ s i n 2 0 ∘ − s i n 1 0 ∘ = tan 1 0 ∘ is proven by simplifying both the numerator and denominator using sine and cosine identities, resulting in both sides being equal. Each step involved careful application of trigonometric formulas to arrive at a valid conclusion. Thus, the original equation is verified to be true.
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Use the sine difference formula to expand sin 2 0 ∘ − sin 1 0 ∘ .
Rewrite the equation to eliminate the fraction.
Apply product-to-sum formulas to simplify the equation further.
Verify the identity by showing that both sides of the equation are equal: 4 1 = 4 1 .
Explanation
Problem Analysis and Objective We are tasked with proving the trigonometric identity: 1 − cos 1 0 ∘ + cos 2 0 ∘ sin 2 0 ∘ − sin 1 0 ∘ = tan 1 0 ∘ Our plan is to simplify the left-hand side (LHS) of the equation and show that it is equal to the right-hand side (RHS), which is tan 1 0 ∘ .
Simplify the Numerator We will use the sine difference formula to simplify the numerator. The formula is: sin A − sin B = 2 cos ( 2 A + B ) sin ( 2 A − B ) In our case, A = 2 0 ∘ and B = 1 0 ∘ . Applying the formula, we get: sin 2 0 ∘ − sin 1 0 ∘ = 2 cos ( 2 2 0 ∘ + 1 0 ∘ ) sin ( 2 2 0 ∘ − 1 0 ∘ ) = 2 cos ( 1 5 ∘ ) sin ( 5 ∘ )
Simplify the Denominator Next, we simplify the denominator. We can rewrite the denominator as: 1 − cos 1 0 ∘ + cos 2 0 ∘ = 1 + ( cos 2 0 ∘ − cos 1 0 ∘ ) Now, we use the cosine difference formula: cos A − cos B = − 2 sin ( 2 A + B ) sin ( 2 A − B ) In our case, A = 2 0 ∘ and B = 1 0 ∘ . Applying the formula, we get: cos 2 0 ∘ − cos 1 0 ∘ = − 2 sin ( 2 2 0 ∘ + 1 0 ∘ ) sin ( 2 2 0 ∘ − 1 0 ∘ ) = − 2 sin ( 1 5 ∘ ) sin ( 5 ∘ ) So, the denominator becomes: 1 − 2 sin ( 1 5 ∘ ) sin ( 5 ∘ )
Substitute and Simplify Now, substitute the simplified expressions for the numerator and denominator into the original equation: 1 − 2 sin ( 1 5 ∘ ) sin ( 5 ∘ ) 2 cos ( 1 5 ∘ ) sin ( 5 ∘ ) This expression does not immediately simplify to tan 1 0 ∘ . Let's try a different approach for the denominator. We can rewrite the denominator using the identity cos ( 2 x ) = 2 cos 2 ( x ) − 1 and cos ( x ) = 1 − 2 sin 2 ( x /2 ) .
1 − cos 1 0 ∘ + cos 2 0 ∘ = 1 − ( 1 − 2 sin 2 5 ∘ ) + ( 2 cos 2 1 0 ∘ − 1 ) = 2 sin 2 5 ∘ + 2 cos 2 1 0 ∘ − 1 This also doesn't seem to lead to a simple expression. Let's go back to the original simplified expression and try to manipulate it to match tan 1 0 ∘ .
Final Proof Let's try another approach. We want to prove that 1 − cos 1 0 ∘ + cos 2 0 ∘ sin 2 0 ∘ − sin 1 0 ∘ = tan 1 0 ∘ which is equivalent to sin 2 0 ∘ − sin 1 0 ∘ = tan 1 0 ∘ ( 1 − cos 1 0 ∘ + cos 2 0 ∘ ) sin 2 0 ∘ − sin 1 0 ∘ = cos 1 0 ∘ sin 1 0 ∘ ( 1 − cos 1 0 ∘ + cos 2 0 ∘ ) cos 1 0 ∘ ( sin 2 0 ∘ − sin 1 0 ∘ ) = sin 1 0 ∘ ( 1 − cos 1 0 ∘ + cos 2 0 ∘ ) cos 1 0 ∘ sin 2 0 ∘ − cos 1 0 ∘ sin 1 0 ∘ = sin 1 0 ∘ − sin 1 0 ∘ cos 1 0 ∘ + sin 1 0 ∘ cos 2 0 ∘ cos 1 0 ∘ sin 2 0 ∘ = sin 1 0 ∘ + sin 1 0 ∘ cos 2 0 ∘ Using the product-to-sum formula, sin A cos B = 2 1 [ sin ( A + B ) + sin ( A − B )] , we have: cos 1 0 ∘ sin 2 0 ∘ = 2 1 [ sin ( 3 0 ∘ ) + sin ( 1 0 ∘ )] = 2 1 [ 2 1 + sin 1 0 ∘ ] sin 1 0 ∘ cos 2 0 ∘ = 2 1 [ sin ( 3 0 ∘ ) + sin ( − 1 0 ∘ )] = 2 1 [ 2 1 − sin 1 0 ∘ ] Substituting these back into the equation: 2 1 [ 2 1 + sin 1 0 ∘ ] = sin 1 0 ∘ + 2 1 [ 2 1 − sin 1 0 ∘ ] 4 1 + 2 1 sin 1 0 ∘ = sin 1 0 ∘ + 4 1 − 2 1 sin 1 0 ∘ 4 1 = 4 1 Since the equation holds true, the original identity is proven.
Conclusion The trigonometric identity 1 − c o s 1 0 ∘ + c o s 2 0 ∘ s i n 2 0 ∘ − s i n 1 0 ∘ = tan 1 0 ∘ is proven by manipulating the equation and using trigonometric identities to show that the left-hand side is equal to the right-hand side.
Examples
Trigonometric identities are fundamental in various fields such as physics, engineering, and computer graphics. For example, in signal processing, trigonometric functions are used to analyze and synthesize signals. Understanding and manipulating trigonometric identities allows engineers to simplify complex expressions and design efficient filters. In computer graphics, these identities are used to perform rotations and transformations of objects in 3D space, enabling realistic rendering and animation.
