Question

Solve:
Cos(2∅-18°)=Tan 54°​

Answer 1

We can use trigonometric identities to simplify both sides of the equation and determine whether they are equal.

First, we can use the identity cos(2θ) = cos^2(θ) - sin^2(θ) to rewrite the left-hand side of the equation:

cos(2∅-18°) = cos^2(∅-9°) - sin^2(∅-9°)

Next, we can use the identity tan(θ) = sin(θ) / cos(θ) to rewrite the right-hand side of the equation:

tan 54° = sin 54° / cos 54°

Now we can substitute these expressions into the original equation and simplify:

cos^2(∅-9°) - sin^2(∅-9°) = sin 54° / cos 54°

Using the identity sin^2(θ) + cos^2(θ) = 1, we can rewrite the left-hand side of the equation as:

cos^2(∅-9°) - sin^2(∅-9°) = cos 2(∅-9°)

So the equation becomes:

cos 2(∅-9°) = sin 54° / cos 54°

Using the identity sin(θ - φ) = sin θ cos φ - cos θ sin φ, we can rewrite the right-hand side of the equation as:

sin 54° / cos 54° = sin(60° - 6°) / cos(60° - 6°) = [sin 60° cos 6° - cos 60° sin 6°] / [cos 60° cos 6° + sin 60° sin 6°] = tan 6°

Therefore, the equation simplifies to:

cos 2(∅-9°) = tan 6°

We can solve for ∅ by using the inverse cosine function on both sides:

2(∅-9°) = cos^-1(tan 6°)

∅-9° = 0.3807

∅ = 9.3807°

So the equation cos(2∅-18°) = Tan 54° is true when ∅ is approximately 9.3807 degrees