11.1 Graphs, trigonometric identities, and solving trigonometric  (Page 9/12)

General solution using periodicity

Up until now we have only solved trigonometric equations where the argument (the bit after the function, e.g. the $\theta$ in $cos\theta$ or the $(2x-7)$ in $tan(2x-7)$ ), has been $\theta$ . If there is anything more complicated than this we need to be a little more careful. Let us try to solve $tan(2x-{10}^{\circ })=2,5$ in the range $-{360}^{\circ }\le x\le {360}^{\circ }$ . We want solutions for positive tangent so using our CAST diagram we know to look in the 1 ${}^{\mathrm{st}}$ and 3 ${}^{\mathrm{rd}}$ quadrants. Our calculator tells us that $arctan(2,5)=68,2^{\circ }$ . This is our reference angle. So to find the general solution we proceed as follows:

This is the general solution. Notice that we added the ${10}^{\circ }$ and divided by 2 only at the end. Notice that we added ${180}^{\circ }·n$ because the tangent has a period of ${180}^{\circ }$ . This is also divided by 2 in the last step to keep the equation balanced. We chose quadrants I and III because $tan$ was positive and we used the formulae $\theta$ in quadrant I and $({180}^{\circ }+\theta )$ in quadrant III. To find solutions where $-{360}^{\circ }<x<{360}^{\circ }$ we substitue integers for $n$ :

• $n=0$ ; $x=39,1^{\circ }$ ; $129,1^{\circ }$
• $n=1$ ; $x=129,1^{\circ }$ ; $219,1^{\circ }$
• $n=2$ ; $x=219,1^{\circ }$ ; $309,1^{\circ }$
• $n=3$ ; $x=309,1^{\circ }$ ; $399,1^{\circ }$ (too big!)
• $n=-1$ ; $x=-50,9^{\circ }$ ; $39,1^{\circ }$
• $n=-2$ ; $x=-140,1^{\circ }$ ; $-50,9^{\circ }$
• $n=-3$ ; $x=-230,9^{\circ }$ ; $-140,9^{\circ }$
• $n=-4$ ; $x=-320,9^{\circ }$ ; $-230,9^{\circ }$

Solution: $x=-320,9^{\circ };-{230}^{\circ };-140,9^{\circ };-50,9^{\circ };39,1^{\circ };129,1^{\circ };219,1^{\circ }$ and $309,1^{\circ }$

Linear trigonometric equations

Just like with regular equations without trigonometric functions, solving trigonometric equations can become a lot more complicated. You should solve these just like normal equations to isolate a single trigonometric ratio. Then you follow the strategy outlined in the previous section.

Quadratic and higher order trigonometric equations

The simplest quadratic trigonometric equation is of the form

This type of equation can be easily solved by rearranging to get a more familiar linear equation