How do you solve trigonometric equations?
To help explain we will use the following example: Show that cos(2x) = 2cos2x - 1. Hence solve the equation cos(2x) + cosx = 0 for 0o < x < 360o. Answer: We know from our sum-difference formulas that cos(u+v) = cos(u)cos(v) - sin(u)sin(v) and in this case u = v = x. Simplifying to cos(2x) = cos2x - sin2x. We also know from our pythagorean identities that sin2x + cos2x = 1, implying that sin2x = 1 - cos2x. We can substitute this into our equation for cos(2x), giving cos(2x) = cos2x - (1 - cos2x). Expanding the brackets gives cos2x - 1 + cos2x. Simplifying gives 2cos2x - 1. This completes the first part of the question. We are given the equation cos(2x) + cosx = 0 we can now substutute in cos(2x) = 2cos2x - 1 which gives 2cos2x - 1 + cosx = 0. Using the quadratic formula we find that cosx = (-1 + 3)/4 = 0.5 or cosx = (-1 - 3)/4 = -1. To find x we use inverse cos: x = cos-1(0.5) = 60o or x = cos-1(-1) = 180o. We can then sketch the cos(x) graph, 0o < x < 360o, find the points where x = 60o and 180o. We then draw vetricle lines across our sketch and find where these verticle lines touch the graph, this finds other solutions within the range, in this case there is one more solution, x = 300o.
Related Maths A Level answers
All answers ▸Find the range of a degree-2 polynomial function such as 2x^2 +1, or x^2 + 2x - 3.
A curve has equation y = f(x) and passes through the point (4,22). Given that f'(x) = 3x^2 - 3x^(1/2) - 7 use intergration to find f(x).
You deposit 500 pounds at time t=0. At t=5 years, you have 800 pounds. The amount of money you have in the bank can be modeled as V(t)=A*(1+r)^t, where r is the interest rate. Find A and the interest rate r. After how many years will you have 1200 pounds.
How do I find dy/dx for a given equation, once this is found how do I find the value of x such that dy/dx = 0.