Consider how the given equation can be modified using all of the various Trigonometric Identities.
No solution.
Practice makes perfect
We want to solve the following equation.
sin ( θ - π/2 ) = sec θ
We will first simplify the left-hand side of the situation. To do so,
we need to consider which Trigonometric Identities will be useful in this equation. Let's recall the Angle Difference Identity for the sine.
sin ( A - B ) = sin A cos B - cos A sin BWith this relationship in mind, let's simplify the left-hand side of the equation.
We have proven the following identity.
sin ( θ - π/2 ) = -cos θ
Let's use it to simplify the left-hand side of our equation.
sin ( θ - π/2 ) &= sec θ
&⇕
-cos θ &= sec θ
Notice that we now have a much simpler equation. Now, let's recall the Secant Identity
sec θ = 1/cos θ
Let's use this identity to express our equation solely in terms of cos θ. However, note that since we have a cos θ in the denominator, we need to check whether for any solutions we might find cos θ is non-zero.
Now, notice that the left-hand side contains a square, while the right-hand side is negative. This means that our equation has no solution, as no real number multiplied by itself becomes negative.
