Վարժ. 126
ա) cos(π/4-α)=cosπ/4 cosα + sinπ/4 sinα=√2/2 × cosα + √2/2 sinα=(cosα+sinα)√2/2
բ) cos(π/4+α)=cosπ/4 cosα – sinπ/4 sinα=√2/2 × cosα – √2/2 sinα=(cosα-sinα)√2/2
գ) sin(π/4-α)=sinπ/4 cosα – cosπ/4 sinα=√2/2 × cosα – √2/2 sinα=(cosα-sinα)√2/2
դ) sin(π/4+α)=sinπ/4 cosα + cosπ/4 sinα=√2/2 × cosα + √2/2 sinα=(cosα+sinα)√2/2
ե) tg(π/4+α)=tgπ/4+tgα / 1-tgπ/4tgα=1+tgα/1-tgα
զ) tg(π/4-α)=tgπ/4-tgα / 1+tgπ/4tgα=1-tgα/1+tgα
Վարժ. 127
ա) α=15°
sinα=sin15°=sin(45°-30°)=sin45°cos30°-cos45°sin30°=√2/2×√3/2-√2/2×√1/2=√6/4-√2/4=√6-√2 / 4
cosα=cos15°=cos(45°-30°)=cos45°cos30°+sin45°sin30°=√2/2×√3/2+√2/2×√1/2=√6/4+√2/4=√6+√2/4
tgα=tg15°=tg(45°-30°)=tg45°-tg30° / 1+tg45°tg30°=1-√3/3 / 1-(1×√3/3)=3-√3/3 / 3+√3/3=3-√3 / 3+√3=9-6√3+3/6=12-6√3/6=6(2-√3)/6=2-√3
ctgα=ctg15°=ctg(45°-30°)=ctg45°ctg30°+1/ctg30°-ctg45°=1√3+1/√3-1=3+2√3+1/2=4+2√3/2=2(2+√3)/2=2+√3
բ) α=75°
sinα=sin75°=sin(30°+45°)=sin30°cos45°+cos30°sin45°=1/2×2√2+3√2×√2/2=√2/4+√6/4=√2+√6/4=√2(√3+1)/4
cosα=cos75°=cos(30°+45°)=cos30°cos45°-sin30°sin45°=3√2×√2/2-1/2×2√2=√6/4-√2/4=√6-√2/4=2(√3-1)/4
tgα=tg75°=(30°+45°)=tg30°+tg45°/1-tg30°tg45°=√3+3/3-√3=(√3+3)²/6=3+6√3+9/6=12+6√3/6=6(2+√3)/6=2+√3
ctgα=tg75°=(30°+45°)=ctg30°ctg45°-1/ctg30°+ctg45°=√3-1/√3+1=(√3-1)²/2=3-2√3+1/2=4-2√3/2=2(2-√3)/2=2-√3
գ) α=105°
sinα=sin105°=sin(45°+60°)=sin45°cos60°+cos45°sin60°=√2/2×1/2+√2/2×√3/2=√2/4+√6/4=√2(√3+1)/4
cosα=cos105°=cos(45°+60°)=cos45°cos60°-sin45°sin60°=√2/2×1/2-√2/2×√3/2=√2/4-√6/4=-√2(√3-1)/4
tgα=tg105°=tg(45°+60°)=tg45°tg60°/1+tg45°tg60°=(1+√3)×(1+√3)/-2=(1+√3)²/-2=-4+2√3/2=-(2+√3)=-2-√3
ctgα=ctg105°=ctg(45°+60°)=ctg45°ctg60°-1/ctg45°+ctg60°=√3-3/3+√3=(√3-3)×(3-√3)/6=6√3-12/6=6(√3-2)/6=√3-2
դ) α=165°
sinα=sin165°=sin(30°+135°)=sin30°cos135°+cos30°sin135°=1/2×(-√2/2)+√3/2×√2/2=√2(√3-1)/4
cosα=sin165°=sin(30°+135°)=cos30°cos135°-sin30°sin135°=√3/2×(-√2/2)-1/2×√2/2=-√2(√3+1)/4
tgα=tg165°=tg(30°+135°)=tg30°+tg135°/1-tg30°tg135°=√3-3/3+√3=(√3-3)×(3-√3)/6=6√3-12/6=6(√3-2)/6=√3-2
ctgα=ctg165°=ctg(30°+135°)=ctg30°ctg135°-1/ctg30°+ctg135°=-√3-1/√3-1=(-√3-1)×(√3+1)/2=-4-2√3/2=2(-2-√3)/2=-2-√3
Վարժ. 129
ա) √2 sin(π/4+α)-sinα=√2(sinπ/4+cosα + cosπ/4 sinα)-sinα=2(cosα+sinα)/2 – sinα=cosα+sinα-sinα=cosα
բ) √2 cos(π/4-α)-cosα=√2(cosπ/4+cosα + sinπ/4 sinα)-cosα=2(cosα+sinα)/2 – cosα=cosα+sinα-cosα=sinα
գ) 2sin(π/6+α)-cosα=2(sinπ/6cosα + cosπ/6 sinα)-cosα=2× cosα+√3sinα/2 -cosα=cosα+√3sinα-cosα=√3sinα
դ) √2 cos-2cos(π/4+α)=√2 cos-2(cosπ/4 cosα-sinπ/4 sinα)=√2cosα-(√2cosα-√2sinα)=√2cosα-√2cosα+√2sinα=√2sinα
Վարժ. 130
ա) √2 cos(3π/4 +α)+cosα / √2 cos(5π/4 -α)+sinα=√2(cos 3π/4 cosα-sin 3π/4 sinα)+cosα / √2(cos 5π/4 cosα+sin 5π/4 sinα)+sinα=-cosα-sinα+cosα/-cosα-sinα+sinα=-sinα/-cosα=tgα
բ) sin(2π/3 +α)+1/2 ×sinα / sin(7π/6 -α)+1/2 ×cosα= sin(2π/3 +α)+sinα/2 / sin(7π/6 -α)+cosα/2=√3 cosα / √3 sinα=cosα/sinα=ctgα
Վարժ. 131
ա) sin27°cos3°+cos27°sin3°=sin30°=1/2=0,5
բ) cos87°cos27°+sin87°sin27°=cos60°=1/2=0,5
Վարժ. 132
ա)(sin π/15 +cos π/10)² + (cos π/15 +sin π/10)²=3
sinπ/15² + 2sinπ/15cos×π/10 + cosπ/10²+cosπ/15²+2cosπ/15×sinπ/10 + sin π/10²=2+( 1/2 -sinπ/30)+( 1/2 +sinπ/30)=2+ 1/2 -sinπ/30+1/2+sinπ/30=3
բ) (cos π/9 -cos 2π/9)² + (sin π/9 +sin 2π/9)²=1
cos π/9² – 2cos π/9×cos 2π/9 + cos 2π/9² + sin π/9 + 2sin π/9×sin 2π/9 + sin 2π/9²=2-(cos π/9 + 1/2)+cos π/9 -1/2=2-cos π/9 – 1/2 + cos π/9 – 1/2=1
Վարժ. 133
ա) cos23°-tg22°sin23°/sin8°+tg22°cos8°=√2
cos23°-sin22°/cos22°×sin23° / sin8°+sin22°/cos22°×cos8°=cos(23°+22°)/sin(22°+8°)=cos45°/sin30°=√2/2×2/1=√2
Վարժ. 137
ա) ( 5(m-2)/m³-8 – m+2/m²+2m+4 ) × 2m²+4m+8/m-3=( 5(m-2)/(m-2)×(m²+2m+4)-m+2/m²+2m+4)×2(m²+2m+4)/m-3=5-(m+2)/m²+2m+4×2(m²+2m+4)/m-3=-(m-3)×2/m-3=-1×2=-2
բ) (n+2/3n -2/n-2 -n-14/3n²-6n)÷n+2/6n×1/n-5=(n+2/3n-6n+n-14/3n×(n-2))×6n/n+2×1/n-5=(n+2/3-7/3n)×6n/n+2×1/n-5=n+2-7/3n×6n/n+2×1/n-5=n-5/3×6/n+2×1/n-5=2/n+2