If cos(t)=–7/9, find the values of the following trigonometric functions.
Note: Give exact answers, do not use decimal numbers. The answer should be a fraction or an arithmetic expression.
a) cos (2t)
b) sin (2t)
c) cos(1/2)
d) sin (1/2)
i don't even know what this question is asking me to find!
I will do one for you. It is asking you to apply the double and half angle formulas.
Cos2t=sin^t -cos^2 t
Now do you know how to get the sint, given cost?
Use pythoregean theorm.
This question is asking you to find the values of certain trigonometric functions given that cos(t) = -7/9. Let's go through each part step-by-step:
a) To find cos(2t), we can use the double-angle identity for cosine: cos(2t) = 2*cos^2(t) - 1.
Substituting cos(t) = -7/9 into the formula, we get:
cos(2t) = 2*(-7/9)^2 - 1
= 2*(49/81) - 1
= 98/81 - 81/81
= (98 - 81)/81
= 17/81
So the value of cos(2t) is 17/81.
b) To find sin(2t), we can use the double-angle identity for sine: sin(2t) = 2*sin(t)*cos(t).
Substituting cos(t) = -7/9 into the formula, we get:
sin(2t) = 2*sin(t)*(-7/9)
= -14/9 * sin(t)
Since we don't have the value of sin(t), we can't find the exact value of sin(2t) without more information.
c) To find cos(1/2), we can use the cosine function. However, the value of cos(1/2) cannot be expressed exactly as a fraction or arithmetic expression. It is an irrational number approximately equal to 0.87758.
d) To find sin(1/2), we can use the sine function. However, the value of sin(1/2) cannot be expressed exactly as a fraction or arithmetic expression. It is an irrational number approximately equal to 0.47943.
This question is asking you to find the values of different trigonometric functions given a specific value for "cos(t)". The given value is "cos(t) = -7/9".
To find the values of the trigonometric functions, we can use some key trigonometric identities and properties. Let's break down each part of the question and explain how to find the values:
a) cos(2t):
The double-angle identity for cosine states: cos(2t) = 2cos^2(t) - 1.
We substitute the given value for cos(t) into the formula:
cos(2t) = 2(cos(t))^2 - 1 = 2(-7/9)^2 - 1 = 2(49/81) - 1 = 98/81 - 1 = (98 - 81)/81 = 17/81.
b) sin(2t):
The double-angle identity for sine states: sin(2t) = 2sin(t)cos(t).
We substitute the given value for cos(t) and use a Pythagorean identity to solve for sin(t):
sin(2t) = 2sin(t)cos(t) = 2sin(t)(-7/9).
Now, we need to find sin(t). The Pythagorean identity states: sin^2(t) + cos^2(t) = 1.
Since we know cos(t) = -7/9, we can find sin(t):
sin^2(t) + (-7/9)^2 = 1
sin^2(t) + 49/81 = 1
sin^2(t) = 1 - 49/81
sin^2(t) = 32/81
sin(t) = sqrt(32/81)
sin(t) = sqrt(32)/9
sin(t) = (4sqrt(2))/9
Now, we can substitute the value of sin(t) into sin(2t):
sin(2t) = 2sin(t)(-7/9) = 2((4sqrt(2))/9)(-7/9) = (8sqrt(2)(-7))/81 = (-56sqrt(2))/81.
c) cos(1/2):
Here, the question is asking for the value of cosine at an angle of 1/2 radians.
To find the exact value of cos(1/2), we can use tables or calculators that provide trigonometric values. On such tables, you can find that cos(1/2) is approximately equal to 0.87758. However, since the question specifies that we should give the answer as a fraction or an arithmetic expression, we leave it as cos(1/2).
d) sin(1/2):
Using a similar approach, we can find the value of sin(1/2) using trigonometric tables or calculators. It is approximately equal to 0.47942. However, we leave it as sin(1/2) as per the question's requirements.
In summary, the values of the trigonometric functions are:
a) cos(2t) = 17/81
b) sin(2t) = (-56sqrt(2))/81
c) cos(1/2) = cos(1/2) (exact value)
d) sin(1/2) = sin(1/2) (exact value)