Question

1) sin^2a+cos^2a-tga×ctga 2)Довести Тотожність:а) (1-sin^2a)×tg^2a=1-cos^2a ;б) tg^2a-sin^2a=tg^2a×sin^2a3) Дано: cosa=0,80<a<п/2Знайти:sinatgactga​

Answer 1

Ответ:

1) sin^2a+cos^2a-tgax*ctga = 1-1 = 0 (т.к sin^2a+cos^2a = 1 и tgax*ctga = 1 )

3) 0<a<п/2 ⇒ 1 четверть

   sina = $\sqrt{1-cos^2a}$ = $\sqrt{1-\frac{64}{100} }$ = $\sqrt{\frac{36}{100} }$ = +-$\frac{6}{10}$ т.к sina в 1 четверти положительный, то sina = $\frac{6}{10}$

   tga= $\frac{sina}{cosa}$ = $\frac{6}{10} / \frac{8}{10}$ = $\frac{6}{10} * \frac{10}{8}$ = $\frac{3}{4}$

   ctga = $\frac{cosa}{sina}$ = $\frac{4}{3}$

   Ответ: sina=$\frac{6}{10}$, tga = $\frac{3}{4}$ , ctga= $\frac{4}{3}$

Answer 2

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Answer 3

1. Let's simplify the expression step by step:

   • We know that sin^2(a) + cos^2(a) = 1 (trigonometric identity).
   • We also know that tan(a) = sin(a) / cos(a) and cot(a) = 1 / tan(a) = cos(a) / sin(a).

   Now, let's substitute the values: sin^2(a) + cos^2(a) - tan(a) * cot(a) = 1 - (sin(a) / cos(a)) * (cos(a) / sin(a)) [Using the trigonometric identities] = 1 - 1 = 0

2. Доведення тотожностей:

   а) Starting with the left side: (1 - sin^2(a)) * tan^2(a) = cos^2(a) * sin^2(a) / cos^2(a) [Using the trigonometric identity tan(a) = sin(a) / cos(a)] = sin^2(a)

   Now, the right side: 1 - cos^2(a) = sin^2(a) [Using the trigonometric identity sin^2(a) + cos^2(a) = 1]

   Since the left side equals the right side, the identity (1 - sin^2(a)) * tan^2(a) = 1 - cos^2(a) is proved.

   б) Starting with the left side: tan^2(a) - sin^2(a) = sin^2(a) / cos^2(a) - sin^2(a) [Using the trigonometric identity tan(a) = sin(a) / cos(a)] = sin^2(a) * (1 / cos^2(a) - 1) = sin^2(a) * (cos^2(a) / cos^2(a) - 1) = sin^2(a) * (1 - cos^2(a) / cos^2(a)) = sin^2(a) * (sin^2(a) / cos^2(a))

   Now, the right side: tan^2(a) * sin^2(a) = (sin^2(a) / cos^2(a)) * sin^2(a) = sin^2(a) * (sin^2(a) / cos^2(a))

   Since the left side equals the right side, the identity tan^2(a) - sin^2(a) = tan^2(a) * sin^2(a) is proved.

3. Дано: cos(a) = 0.80, 0 < a < π/2 Треба знайти: sin(a), tan(a), cot(a)

   Знаючи cos(a), ми можемо використовувати тригонометричні тотожності, щоб знайти інші значення:

   З тотожності sin^2(a) + cos^2(a) = 1: sin^2(a) = 1 - cos^2(a) sin(a) = √(1 - cos^2(a))

   З тотожності tan(a) = sin(a) / cos(a): tan(a) = sin(a) / cos(a)

   З тотожності cot(a) = 1 / tan(a): cot(a) = 1 / tan(a)

   Підставимо значення cos(a) = 0.80 у вирази, щоб знайти sin(a), tan(a) і cot(a).