42: Harder Trig Equations “Teach A Level Maths” Vol. 1: AS Core Modules 42: Harder Trig Equations © Christine Crisp

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1st solution: e.g.1 Solve the equation for the interval Solution: Let so, We can already solve this equation BUT the interval for x is not the same as for . There will be 4 solutions ( 2 for each cycle ). 1st solution: ( Once we have 2 adjacent solutions we can add or subtract to get the others. ) Sketch to find the 2nd solution:

for So, For , the other solutions are So, N.B. We must get all the solutions for x before we find . Alternate solutions for are NOT apart.

We can use the same method for any function of . e.g. (a) for Use and e.g. (b) for Use and e.g. (c) for Use and

SUMMARY Solving Harder Trig Equations Replace the function of by x. Write down the interval for solutions for x. Find all the solutions for x in the required interval. Convert the answers to values of .

Exercise Solve the equation for Solution: Let Principal value: So, 1 -1 So,

We sometimes need to give answers in radians We sometimes need to give answers in radians. If so, we may be asked for exact fractions of . e.g. Principal value is Tip: If you don’t remember the fractions of , use your calculator in degrees and then convert to radians using radians So, from the calculator rads. If an exact value is not required, then switch the calculator to radian mode and get (3 d.p.)

e.g. 2 Solve the equation giving exact answers in the interval . The use of always indicates radians. Solution: Let ( or ) 1st solution is For “tan” equations we usually keep adding to find more solutions, but working in radians we must remember to add . So,

e.g. 3 Solve the equation for the interval . Solution: Let Principal value: rads. Sketch for a 2nd value:

for 1 -1 2nd value: repeats every , so we add to the principal value to find the 3rd solution: Ans:

Solution: We can’t let so we use a capital X e.g. 4 Solve the equation for giving the answers correct to 2 decimal places. Solution: We can’t let so we use a capital X ( or any another letter ). Let so 2 1 We need to use radians but don’t need exact answers, so we switch the calculator to radian mode. Principal value: Sketch for the 1st solution that is in the interval:

for 1 -1 1st solution is 2nd solution is ( 2 d.p.) Multiply by 2: Ans:

Exercise Solve the equation for giving the answers as exact fractions of . 2. Solve the equation for giving answers correct to 1 decimal place.

Solutions Solve the equation for Solution: Let Principal value: Add :

Þ Solutions 2. Solve the equation for giving answers correct to 1 decimal place. Þ Solution: Let Principal value: Sketch for the 2nd solution:

The 2nd value is too large, so we subtract for 1 -1 The 2nd value is too large, so we subtract Add : Ans:

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SUMMARY Replace the function of by x. Solving Harder Trig Equations Write down the interval for solutions for x. Find all the solutions for x in the required interval. Convert the answers to values of .

1st solution: e.g. 1 Solve the equation for the interval Sketch to find the 2nd solution: Solution: Let so, ( Once we have 2 adjacent solutions we can add or subtract to get the others. ) There will be 4 solutions ( 2 for each cycle ). We can already solve this equation BUT the interval for x is not the same as for .

So, For , the other solutions are N.B. We must get all the solutions for x before we find . Alternate solutions for are NOT apart. for

e.g. (a) for Use and We can use the same method for any function of . e.g. (b) for Use and e.g. (c) for

The use of always indicates radians. e.g. 2 Solve the equation giving exact answers in the interval . Solution: Let ( or ) 1st solution is For “tan” equations we usually keep adding to find more solutions, but working in radians we must remember to add .

Solution: Let e.g. 3 Solve the equation for the interval . Principal value: rads. Sketch for a 2nd solution:

2nd value: repeats every , so we add to the 1st value: for Ans: So,

Solution: We can’t let so we use a capital X e.g. 4 Solve the equation for giving the answers correct to 2 decimal places. We need to use radians but don’t need exact answers, so we switch the calculator to radian mode. Solution: We can’t let so we use a capital X ( or any another letter ). Let so Principal value: Sketch for 1st solution that is in the interval: 2 1

1st solution is 2nd solution is Multiply by 2: Ans: for ( 2 d.p.)