Question

Oblicz wartości pozostałych funkcji trygometrycznych kąta ostrego alfa jeśli wiadomo że.Zadanie w załączniku.

Answer 1

Odpowiedź:

a)

sinα = 0,1

sin²α = (0,1)² = 0,01

1 - cos²α = 0,01

cos²α = 1 - 0,01 = 0,99

cosα = √0,99 = √(99/100) = √99/10 = √(9 * 11)/10 = 3√11/10

tgα = sinα/cosα = 0,1 : 3√11/10 = 1/10 * 10/3√11 = 1/3√11 = √11/(3 * 11) =

= √11/33

ctgα = 1/tgα = 33/√11 = 33√11/11 = 3√11

b)

cosα = 0,9

cos²α = (0,9)² = 0,81

1 - sin²α = 0,81

sin²α = 1 - 0,81 = 0,19

sinα = √0,19 = √(19/100) = √19/10

tgα = sinα/cosα = √19/10 : 9/10 = √19/10 * 10/9 = √19/9

ctgα = 1/tgα = 9/√19 = 9√19/19

c)

cosα = 1/6

cos²α = (1/6)² = 1/36

1 - sin²α = 1/36

sin²α = 1 - 1/36 = 35/36

sinα = √(35/36) = √35/6

tgα = sinα/cosα = √35/6 : 1/6 = √35/6 * 6 = √35

ctgα = 1/tgα = 1/√35 = √35/35

d)

sinα = √3/3

sin²α = (√3/3)² = 3/9 = 1/3

1 - cos²α = 1/3

cos²α = 1 - 1/3 = 2/3

cosα = √(2/3) = √2/√3 = √2 * √3/3 = √6/3

tgα = sinα/cosα = √3/3 : √6/3 = √3/3 * 3/√6 = √3/√6 = √(3/6) = √(1/2) =

= 1/√2 = √2/2

ctgα = 1/tgα = 2/√2 = 2√2/2 = √2

e)

tgα =  8/15

tg²α = (8/15)² = 64/2258

sin²α/cos²α = 64/225

225sin²α = 64cos²α = 64(1 - sin²α) = 64 - 64sin²α

225sin²α + 64sin²α = 64

289sin²α = 64

sin²α = 64/289

sinα = √(64/289) = 8/17

tgα = sinα/cosα = 8/15

sinα = 8/15 * cosα

8/17 = 8/15 * cosα

cosα = 8/17 : 8/15 = 8/17 * 15/8 = 1/17 * 15 = 15/17

ctgα = 1/tgα = 15/8 = 1 7/8

f)

tgα = √5

tg²α = (√5)² = 5

sin²α/cos²α = 5

sin²α = 5cos²α = 5(1 - sin²α) = 5 - 5sin²α

sin²α + 5sin²α = 5

6sin²α = 5

sin²α = 5/6

sinα = √(5/6) = √5/√6 = √5 * √6/6 = √30/6

cosα = √(1 - sin²α) = √(1 - 30/36) = √(6/36) = √(1/6) = √1/√6 = 1/√6 = √6/6

ctgα = 1/tgα = 1/√5 = √5/5

g)

tgα = 1/4

tg²α = sin²α/cos²α = (1/4)² = 1/16

sin²α = 1/16 * cos²α = 1/16(1 - sin²α) = 1/16 - 1/16sin²α

sin²α + 1/16sin²α = 1/16

(1 1/16)sin²α = 1/16

sin²α = 1/16 : 1 1/16 = 1/16 : 17/16 = 1/16 * 16/17 = 1/17

cosα = √(1 - sin²α) = √(1 - 1/17) = √(16/17) = 4/√17 = 4√17/17

ctgα = 1/tgα = 4

h)

tgα = 2√3

tg²α = sin²α/cos²α = (2√3)² = 4 * 3 = 12

sin²α = 12cos²α = 12(1 - sin²α) = 12 - 12sin²α

sin²α + 12sin²α = 12

13sin²α = 12

sin²α = 12/13

sinα = √(12/13)= √12/√13 = √12 * √13/13 = √156/13 = √(4 * 39)/13 =

= 2√39/13

cosα = √(1 - sin²α) = √(1 - 12/13) = √(1/13) = 1/√13 = √13/13

ctgα =1/tgα = 1/2√3 = √3/(2 * 3) = √3/6