Question

Найти sin a, tg a, ctg a, если cos a= -84/85, П/2<а<П

Answer 1

Ответ в файле

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Answer 2

To find the values of sin a, tan a, and cot a, given that cos a = -84/85 and Π/2 < a < Π, we can use the Pythagorean trigonometric identity and the definitions of tangent and cotangent.

Let's start by finding sin a:

Step 1: Use the Pythagorean trigonometric identity: sin^2(a) + cos^2(a) = 1 Step 2: Substitute the value of cos a: sin^2(a) + (-84/85)^2 = 1 Step 3: Solve for sin a: sin^2(a) = 1 - (-84/85)^2 Step 4: Take the positive square root (because a is in the second quadrant): sin a = √(1 - (-84/85)^2)

Next, let's find tan a:

Step 5: Use the definition of tangent: tan a = sin a / cos a Step 6: Substitute the values of sin a and cos a: tan a = √(1 - (-84/85)^2) / (-84/85)

Finally, let's find cot a:

Step 7: Use the definition of cotangent: cot a = 1 / tan a Step 8: Substitute the value of tan a: cot a = 1 / (√(1 - (-84/85)^2) / (-84/85)) Step 9: To simplify, multiply the numerator and denominator by (-85): cot a = (-85) / √(1 - (-84/85)^2)

Now, let's calculate the values:

Step 4: sin a = √(1 - (-84/85)^2) sin a ≈ √(1 - (7056/7225)) sin a ≈ √(169/7225) sin a ≈ √(1/45) sin a ≈ 1/√45 sin a ≈ √45/45 sin a ≈ √5/5 sin a ≈ √5/5 * √5/√5 (Rationalize the denominator) sin a ≈ √25/5 sin a ≈ 5/5 sin a = 1

Step 6: tan a = √(1 - (-84/85)^2) / (-84/85) tan a ≈ 1 / (-84/85) tan a ≈ -85/84

Step 9: cot a = (-85) / √(1 - (-84/85)^2) cot a ≈ -85 / (1/√45) cot a ≈ -85 * √45 cot a ≈ -85 * √(9*5) cot a ≈ -85 * 3 * √5 cot a ≈ -255 * √5

So, the values are: sin a = 1, tan a ≈ -85/84, cot a ≈ -255 * √5.