Find the following values (while justifying all steps):
(a) [tex]$\cos \left(765^{\circ}\right)$[/tex]
(b) [tex]$\cot \left(-\frac{7 \pi}{3}\right)$[/tex]
Answer (2)
Find the coterminal angle for 76 5 ∘ which is 4 5 ∘ , then calculate cos ( 4 5 ∘ ) .
Find the coterminal angle for − 3 7 π which is − 3 π , then calculate cot ( − 3 π ) .
cos ( 4 5 ∘ ) = 2 2 .
cot ( − 3 7 π ) = − 3 3 .
cos ( 76 5 ∘ ) = 2 2 , cot ( − 3 7 π ) = − 3 3
Explanation
Problem Analysis We are asked to find the values of cos ( 76 5 ∘ ) and cot ( − 3 7 π ) . We will use trigonometric identities to simplify these expressions.
Finding cos(765 degrees) (a) To find cos ( 76 5 ∘ ) , we first need to find an angle θ between 0 ∘ and 36 0 ∘ that is coterminal with 76 5 ∘ . We can do this by subtracting multiples of 36 0 ∘ from 76 5 ∘ until we get an angle in the desired range.
76 5 ∘ − 36 0 ∘ = 40 5 ∘
40 5 ∘ − 36 0 ∘ = 4 5 ∘
So, 76 5 ∘ is coterminal with 4 5 ∘ . Therefore, cos ( 76 5 ∘ ) = cos ( 4 5 ∘ ) .
We know that cos ( 4 5 ∘ ) = 2 2 ≈ 0.7071 .
Finding cot(-7pi/3) (b) To find cot ( − 3 7 π ) , we can use the identity cot ( − θ ) = − cot ( θ ) . So, cot ( − 3 7 π ) = − cot ( 3 7 π ) .
Now, we need to find an angle ϕ between 0 and 2 π that is coterminal with 3 7 π . We can do this by subtracting multiples of 2 π from 3 7 π until we get an angle in the desired range.
3 7 π − 2 π = 3 7 π − 3 6 π = 3 π
So, 3 7 π is coterminal with 3 π . Therefore, cot ( 3 7 π ) = cot ( 3 π ) .
We know that cot ( 3 π ) = s i n ( 3 π ) c o s ( 3 π ) = 2 3 2 1 = 3 1 = 3 3 ≈ 0.5774 .
Thus, cot ( − 3 7 π ) = − cot ( 3 7 π ) = − 3 3 ≈ − 0.5774 .
Final Answer Therefore, the values are:
(a) cos ( 76 5 ∘ ) = 2 2
(b) cot ( − 3 7 π ) = − 3 3
Examples
Understanding trigonometric functions like cosine and cotangent is crucial in fields like physics and engineering. For instance, when analyzing the motion of a pendulum, the cosine function describes the horizontal displacement of the pendulum bob over time. Similarly, in electrical engineering, the cotangent function can appear when dealing with impedance in AC circuits. Mastering these concepts allows for accurate modeling and prediction of real-world phenomena.
The value of cos ( 76 5 ∘ ) is 2 2 and the value of cot ( − 3 7 π ) is − 3 3 .
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