SOLVING INVERSE TRIG FUNCTIONS WITH SINE COSINE AND TAN

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Prove that

Problem 1 :

cos-1 45 + cos-1 1213 =cos-1 3365

Solution :

Given, cos-1 45 + cos-1 1213 =cos-1 3365cos-1 x + cos-1 y =cos-1 xy - 1 - x2 1 - y2 =cos-1 451213 - 1 - 452 1 - 12132= cos-14865 - 1 - 16251 - 144169= cos-14865 - 25 - 1625169 - 144169= cos-14865 - 92525169= cos-14865 - 35 · 513= cos-14865 -1565= cos-13365 cos-1 45 + cos-1 1213 =cos-1 3365

Hence proved.

Problem 2 :

cos-1 1213 + sin-1 35 =sin-1 5665

Solution :

Given, cos-1 1213 + sin-1 35 =sin-1 5665= sin-1 1 - 12132+ sin-1 35= sin-1 1 - 144169 + sin-1 35= sin-1 169 - 144169 + sin-1 35= sin-1 25169 + sin-1 35= sin-1 513 + sin-1 35sin-1 x + sin-1 y =sin-1 x 1 - y2 + y1 - x2 = sin-1 513 1 - 352 + 351 - 5132= sin-1513 1 - 925 + 351 - 25169= sin-1513 25 - 925 + 35169 - 25169= sin-1513 1625 + 35144169= sin-1513 × 45 + 35 × 1213= sin-12065 + 3665= sin-15665cos-1 1213 + sin-1 35 =sin-1 5665Hence proved.

Problem 3 :

Show that 

sin-1 1213 + cos-1 45 + tan-1 6316 =𝜋

Solution :

Given, sin-1 1213 + cos-1 45 + tan-1 6316= tan-112131 - 12132 + tan-11 - 45245 + tan-1 6316= tan-112131 - 144169 + tan-11 - 162545 + tan-1 6316= tan-11213 169 - 144169 + tan-125 - 162545 + tan-1 6316= tan-1121325169 + tan-192545 + tan-1 6316= tan-11213513 + tan-13545 + tan-1 6316= tan-1 1213 × 135 + tan-1 35 × 54 + tan-1 6316= tan-1 125 + tan-1 34 + tan-1 6316tan-1 x + tan-1 y =𝜋 + tan-1 x + y1 - xy= 𝜋 + tan-1 125 + 341 - 12534 + tan-1 6316= 𝜋 + tan-1 48 + 1520 1 - 3620 + tan-1 6316= 𝜋 + tan-1 6320 20 - 3620 + tan-1 6316= 𝜋 + tan-1 6320 -1420 + tan-1 6316= 𝜋 - tan-1 6320 × 2014 + tan-1 6316= 𝜋 - tan-1 6316 + tan-1 6316= 𝜋 Hence proved.

Problem 4 :

Prove that 

sin-1 35 - sin-1 817 = cos-1 8485

Solution :

Given, sin-1 35 - sin-1 817 = cos-1 8485sin-1 x - sin-1 y =sin-1 x 1 - y2 - y1 - x2 = sin-1 35 1 - 8172 - 8171 - 352= sin-135 1 - 64289 - 8171 - 925= sin-135 289 - 64289 - 81725 - 925= sin-135 225289 - 8171625= sin-135 × 1517 - 817 × 45= sin-1917 - 3285= sin-11385= cos-11 - 13852= cos-11 - 1697225= cos-17225 - 1697225= cos-170567225= cos-1 8485sin-1 35 - sin-1 817 = cos-1 8485Hence proved.

Problem 5 :

Solve

cos-1 x + sin-1 x2 = 𝜋6
Given, cos-1 x + sin-1 x2 = 𝜋6cos-1x + sin-1x = 𝜋2𝜋2 - sin-1x + sin-1 x2 = 𝜋6𝜋2 - 𝜋6 = sin-1x - sin-1 x2 sin-1x - sin-1 x2 = 𝜋3 sin-1x - sin-1 x2 = sin-132sin-1x = sin-132 + sin-1 x2sin-1x = sin-1321 - x24 + x2 1 - 34sin-1x = sin-1324 - x24 + x2 4 - 34sin-1x = sin-132 × 4 - x22 + x2 × 12x = 34 - x24 + x4x - x4 = 34 - x243x4 = 34 - x243x = 34 - x2Squaring on both sides.9x2 = 34 - x2Dividing 3 on each sides.3x2 = 4 - x23x2 + x2 = 44x2 = 4Dividing 4 on each sides.x2 = 1x = ±1

Problem 6 :

If sin-1 2a1 + a2 - cos-1 1 - b21 + b2 = tan-1 2x1 - x2Then prove that x = a - b1 + ab

Solution :

Given, sin-1 2a1 + a2 - cos-1 1 - b21 + b2 = tan-1 2x1 - x22 tan-1 a - 2 tan-1 b = 2 tan-1 xDividing 2 on each sides.tan-1 a - tan-1 b = tan-1 xtan-1a - b1 + ab = tan-1xx = a - b1 + ab

Hence proved.

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