a) cos α=√7/3 0 st<α<90 st sin²α+cos²α=1 sin²α+(√7/3)²=1 sin²α+7/9=1 sin²α=2/9 sinα=√2/3 ==================== tgα=sinα/cosα tgα=(√2/3):(√7/3) tgα=(√2/3)·(3/√7) tgα=√2/√7 tgα=√14/7 ===================================== ctgα=1/tgα ctgα=√7/√2 ctgα=√14/2 ============================================================= b) sinβ=7/8 sin²β+cos²β=1 49/64+cos²β=1 cos²β=15/64 cosβ=√15/8 ====================================== tgβ=sinβ/cosβ tgβ=7/8:√15/8 tgβ=7/√15 tgβ=(7√15)/15 ======================= ctgβ=1/tgβ ctgβ=1:7/√15 ctgβ=√15/7 ============================================================= c) tgy=1/2 tutaj tworzymy uklad rownan {siny/cosy=1/2 {sin²y+cos²y=1 {cosy=2siny podstawiamy do 2. rownania sin²y+4sin²y=1 5sin²y=1 sin²y=1/5 siny=√5/5 ==================== cosγ=2sin γ cosγ=(2√5)/5 ============================ ctgy=1/tgy ctgy=2 ====================================================================== d) ctgγ=8 tgγ=1/8 tworzymy uklad rownan {cosγ/sinγ=8 {sin²γ+cos²γ=1 {cosγ=8sinγ podstawiamy do 2. rown. {sin²γ+64sin²γ=1 65sin²γ=1 sin²γ=1/65 sinγ=√65/65 ======================== cosγ=(8√65)/65 ===================================================================
a) α < 90 cosα = √7/3 sin²α + cos²α = 1 sin²α = 1 - cos²α = 1-(√7/3)² = 1-7/9 = 9/9 - 7/9 = 2/9 sinα = √(2/9) = √2/3 tgα = siα/cosα = √2/3 : √7/3 = √(2/7) tgα * ctgα = 1 ctgα = 1/√(2/7) *√(2/7)/√(2/7 = √(2/7)/(2/7) = 7√(2/7)/2 = 3,5√(2/7) b) sinβ = 7/8 cos²β = 1-sin²β = 1-(7/8)² = 1- 49/64 = 64/64 - 49/64 = 15/64 cosα = √(15/64) = √15/8 tgβ = sinβ/cosβ = 7/8 : √15/8 = 7/8 * 8/√15 = 7/√15 * √15/√15 = 7√15/15 ctgβ = 15/7√15 * √15/√15 = √15/7 c) tgγ = 1/2 ctgγ = 1: 1/2 = 2 b = 1, a = 2 c² = a²+b² = 2²+1² = 5 c = √5 sinγ = b/c = 1/√5 * √5/√5 = √5/5 cosγ = a/c = 2/√5 * √5/√5 = 2√5/5 d) ctgδ = 8 tgδ = 1/8 a = 8, b = 1 c² = a²+b² = 8²+1² = 64+1 = 65 c = √65 sinδ = b/c = 1/√65 *√65/√65 = √65/65 cosδ = a/c = 8/√65 *√65/√65 = 8√65/65
