Problem 85 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. $$\sin 60^{\circ} \csc 60^{\circ}=1$$

Step-by-Step Solution

Answer

The statement \(\sin 60^{\circ} \csc 60^{\circ} = 1\) is true. The reason is that \(\csc x\) is the reciprocal of \(\sin x\). Therefore, their product is always 1.

Step 1: Know your trigonometric identities

As stated earlier, it is important to understand the trigonometric functions used in the equation. For any angle \(x\), \(\csc x = \frac{1}{\sin x}\). This is an important identity to use for verifying the equation.

Step 2: Substituting the identity

Replace \(\csc 60^{\circ}\) with its equivalent using the identity stated in Step 1. Therefore, the equation \(\sin 60^{\circ} \csc 60^{\circ} = 1\) becomes \(\sin 60^{\circ} (\frac{1}{\sin 60^{\circ}}) = 1\).

Step 3: Simplifying the equation

Simplify the equation by performing the multiplication on the left side of the equation. \(\sin 60^{\circ} \) multiplied by \(\frac{1}{\sin 60^{\circ}}\) gives us 1.