If tan 𝜃 = 7/9 and cot 𝜙 = 14/15 , with 𝜃 and 𝜙 in the first quadrant, find the exact value of sin𝜃 − 𝜙.

Answer 1

sin(A-B) = sinAcosB - sinBcosA

tanA = 7/9 = sinA/cosA

cotB = 14/15 = cosB/sinA

sinA = 7/(sqr(49+81)) = 7/sq131

cosA = 9/sqr131

sinB = 15/sqr(14^2+15^2) = 15/sqr421

cosB = 14/sqr421

sin(A-B) = sinAcosB - sinBcosA

= (7/sqr131)(14/sqr421) - (15/sqr421)(9/sqr131)

= (98 - 135)/sqr(131x421) = -37/sqr55151

= about - 0.1576= about -0.16

check the answer, calculate A and B

A = about 37.7

B = about 47

A-B = about -9.3 degrees or 350.7 degrees

sin(A-B) = about -0.16