If [tex]$\cos 45^{\circ}=\frac{1}{\sqrt{2}}$[/tex], prove that [tex]$\tan 22 \frac{1}{2}^{\circ}=\sqrt{2}-1$[/tex]
Answer (1)
Use the half-angle formula for tangent: tan ( 2 x ) = s i n x 1 − c o s x .
Substitute x = 4 5 ∘ into the formula: tan 22 2 1 ∘ = s i n 4 5 ∘ 1 − c o s 4 5 ∘ .
Use the given value cos 4 5 ∘ = 2 1 and the fact that sin 4 5 ∘ = 2 1 .
Simplify the expression to obtain the final result: tan 22 2 1 ∘ = 2 − 1 .
tan 22 2 1 ∘ = 2 − 1
Explanation
State the given information and the goal. We are given that cos 4 5 ∘ = 2 1 and we want to prove that tan 22 2 1 ∘ = 2 − 1 .
Apply the half-angle formula. We will use the half-angle formula for tangent, which is given by: tan ( 2 x ) = sin x 1 − cos x In our case, x = 4 5 ∘ , so we want to find tan 22 2 1 ∘ = tan ( 2 4 5 ∘ ) .
Substitute the values of cos and sin. Substitute x = 4 5 ∘ into the half-angle formula: tan 22 2 1 ∘ = sin 4 5 ∘ 1 − cos 4 5 ∘ We know that cos 4 5 ∘ = 2 1 . Since sin 4 5 ∘ = cos 4 5 ∘ , we also have sin 4 5 ∘ = 2 1 .
Simplify the expression. Substitute the values of cos 4 5 ∘ and sin 4 5 ∘ into the equation: tan 22 2 1 ∘ = 2 1 1 − 2 1 To simplify this expression, we can multiply both the numerator and the denominator by 2 :
tan 22 2 1 ∘ = 2 ( 2 1 ) 2 ( 1 − 2 1 ) = 1 2 − 1 = 2 − 1
Conclusion. Therefore, we have shown that tan 22 2 1 ∘ = 2 − 1 .
Examples
Understanding trigonometric identities like this is useful in various fields such as physics and engineering. For example, when analyzing the motion of a projectile, you might need to calculate the angle at which it should be launched to reach a certain distance. This often involves using trigonometric functions and their relationships to solve for the required angle. In this case, knowing the value of tan 22. 5 ∘ can help simplify calculations and find the solution more efficiently.
