Solving Trig Equations

Pre-Calculus · Axiom Academy

LESSON Solving Trigonometric Equations Isolate the trig function, read off a reference angle, then let the unit circle hand you every solution in the interval. Before anything trigonometric happens, the equation has to read . Our equation is , so we strip the coefficient by dividing both sides by 2 . The trig function ends up alone, equal to a clean ratio. Divide both sides by the coefficient Now it's "trig function = ratio" 2. Take the Inverse for a Reference Angle Now ask: at what angle does sine equal ? The inverse sine answers it — . This is the reference angle : the acute angle whose sine has the right size . It's the seed for every real solution. The reference angle (a number you can recognize) Draw the height across the sine wave; the inverse picks the first place the curve reaches it. It's only the acute base angle. The interval may hold several angles that share this same reference. map to — memorize them and skip the calculator. only ever returns an angle in — that's why it gives the reference, not the full answer. 3. The Unit Circle Gives Every Solution On the unit circle, sine is the height . The horizontal line slices the circle at two points — and sine is positive in exactly two quadrants, Q1 and Q2. The reference angle lands the first; its mirror lands the second. The rule of thumb: a ratio gives two angles per turn — the reference angle, and its partner in the second quadrant where that function keeps the same sign. , where the line first meets the circle ( ).

This is the written version of the interactive lesson above. See the full Pre-Calculus course.