Question 10:
(a)(i) Prove tanA2=1−cosAsinA.(ii) Hence, without using calculator, find the value of tan15o. State your answer in the form p−√q , where p and q are constants.(b)(i) Sketch the graph of y=−32sinA for 0≤A≤2π.(ii) Hence, using the same axes, sketch a suitable straight line to find the number of solutions for the equation (cotA2)(1−cosA)=−A2π for 0≤A≤2π. State the number solutions.
Solution:
(a)(i)
Right hand side,1−cosAsinA=2sin2A22sinA2cosA2 =sinA2cosA2 =tanA2 =(Left hand side)
(a)(ii)
tan(302)=1−cos30osin30otan15o=1−√3212 =2−√3212 =2−√3
(b)(i)


(b)(ii)
(cotA2)(1−cosA)=−A2π1tanA2(1−cosA)=−A2π(sinA1−cosA)(1−cosA)=−A2πsinA=−A2π−32sinA=−A2π×(−32)y=3A4π

Number of solutions = 3
(a)(i) Prove tanA2=1−cosAsinA.(ii) Hence, without using calculator, find the value of tan15o. State your answer in the form p−√q , where p and q are constants.(b)(i) Sketch the graph of y=−32sinA for 0≤A≤2π.(ii) Hence, using the same axes, sketch a suitable straight line to find the number of solutions for the equation (cotA2)(1−cosA)=−A2π for 0≤A≤2π. State the number solutions.
Solution:
(a)(i)
Right hand side,1−cosAsinA=2sin2A22sinA2cosA2 =sinA2cosA2 =tanA2 =(Left hand side)
(a)(ii)
tan(302)=1−cos30osin30otan15o=1−√3212 =2−√3212 =2−√3
(b)(i)


(b)(ii)
(cotA2)(1−cosA)=−A2π1tanA2(1−cosA)=−A2π(sinA1−cosA)(1−cosA)=−A2πsinA=−A2π−32sinA=−A2π×(−32)y=3A4π

Number of solutions = 3