Use the double-angle identity before solving

Show that tan⁡x+cot⁡x>4\tan x+\cot x>4 for 0<x<π/20<x<\pi/2 simplifies to 0<sin⁡2x<1/20<\sin2x<1/2. marks Hence solve it exactly in terms of pi. marks

[2018/DHS/Prelim/I/4]

Total marks: 6

Solution

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Trigonometric identities

Show that tan⁡x+cot⁡x>4\tan x+\cot x>4 for 0<x<π20<x<\frac\pi2 simplifies to 0<sin⁡2x<120<\sin2x<\frac12.

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Hint

  1. Write everything in terms of sin⁡x\sin x and cos⁡x\cos x over a common denominator.

  2. Use sin⁡2x+cos⁡2x=1\sin^2x+\cos^2x=1 and 2sin⁡xcos⁡x=sin⁡2x2\sin x\cos x=\sin2x.

  3. Remember sin⁡2x>0\sin2x>0 on this interval.

Answer · Show

0<sin⁡2x<1/20<\sin2x<1/2.

Explore the idea

tan x + cot x equals 2/sin 2x

The curve is U-shaped on (0, π/2), with its minimum 2 at x = π/4. It crosses y = 4 where sin 2x = ½.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • Tan x + cot x
  • Y = 4
  • Where the curve meets 4
Ideas to try
The curve is above 4 near both ends. Which two intervals does that give?
Solve

Solving trigonometric inequalities

Hence solve tan⁡x+cot⁡x>4\tan x+\cot x>4 exactly in terms of π\pi.

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Hint

  1. Sketch y=sin⁡2xy=\sin2x on 0<x<π20<x<\frac\pi2 with the lines y=0y=0 and y=12y=\frac12.

  2. Find the xx-values where sin⁡2x=12\sin2x=\frac12.

  3. Read off where the curve is between the two lines.

Answer · Solve

0<x<π/120<x<\pi/12 or 5π/12<x<π/25\pi/12<x<\pi/2.

Explore the idea

Where is sin 2x between 0 and ½?

The sine hump is below the line y = ½ near each end.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • Y = sin 2x
  • Y = ½
  • Boundary values
Ideas to try
Which interval does the peak of sin⁡2x\sin2x lie in? Is it a solution?