Exercise 11 Page 915

Rewrite the left-hand side of the identity using the Double-Angle Identities and the definitions of the trigonometric functions to arrive at the right-hand side.

See solution.

Practice makes perfect

We are given right triangle LMN.

We want to verify the following identity using the given triangle. sin 2N = 2n l/m^2To do so, we will rewrite the left-hand side of the identity using the Double-Angle Identities and the definitions of the trigonometric functions to arrive at the right-hand side. Let's start by recalling the Double Angle Identity that involves sine. sin 2 θ = 2sin θ cos θ We will use this identity to rewrite left-hand side of the equation. Note that in our case θ = N.

sin 2N ? = 2n l/m^2

sin(2θ)=2sin(θ)cos(θ)

2sin N cos N ? = 2n l/m^2

Now let's recall the definitions of sine and cosine in right triangles. sin θ = opp/hyp cos θ = adj/hyp We can write these formulas for angle N. Let's highlight the opposite side, the adjacent side, and the hypotenuse in the diagram.

Next, we will substitute the variables from the diagram into the formulas. sin N = n/m cos N = l/m Now we can substitute n m for sin N and l m for cos N in our expression on the left-hand side. Then, we will simplify it.

2sin N cos N ? = 2n l/m^2

SubstituteII

sin N= n/m, cos N= l/m

2* n/m * l/m ? = 2n l/m^2

▼

Evaluate left-hand side

MoveLeftFacToNum

a*b/c= a* b/c

2n/m * l/m ? = 2n l/m^2

MultFrac

Multiply fractions

2n l/m^2 = 2n l/m^2 ✓