If [tex]$\cot _\theta=\frac{3}{4}$[/tex] and the terminal point determined by [tex]$\theta$[/tex] is in quadrant 3, then:
A. [tex]$\csc \theta=-\frac{5}{3}$[/tex]
B. [tex]$\sin \theta=\frac{3}{5}$[/tex]
C. [tex]$\cos \theta=-\frac{3}{5}$[/tex]
D. [tex]$\tan \theta=\frac{4}{3}$[/tex]
Answer (1)
Given cot θ = 4 3 in quadrant 3, determine the correct trigonometric value.
Calculate r = x 2 + y 2 = 5 using x = − 3 and y = − 4 .
Compute cos θ = r x = − 5 3 and tan θ = x y = 3 4 .
Identify the correct option: tan θ = 3 4 .
Explanation
Analyze the given information We are given that cot θ = 4 3 and that the terminal point of θ is in quadrant 3. This means that both the x and y coordinates of the point are negative. We can think of cot θ as y x = 4 3 . Since both x and y are negative, we can set x = − 3 and y = − 4 .
Calculate r Now we need to find the value of r , which is the distance from the origin to the point ( − 3 , − 4 ) . We can use the Pythagorean theorem to find r : r = x 2 + y 2 = ( − 3 ) 2 + ( − 4 ) 2 = 9 + 16 = 25 = 5
Compute trigonometric values Now we can find the values of the trigonometric functions in the given options:
A. csc θ = y r = − 4 5 = − 4 5 = − 1.25 . This does not match option A, which is − 3 5 .
B. sin θ = r y = 5 − 4 = − 5 4 = − 0.8 . This does not match option B, which is 5 3 .
C. cos θ = r x = 5 − 3 = − 5 3 = − 0.6 . This matches option C.
D. tan θ = x y = − 3 − 4 = 3 4 = 1.333... . This matches option D.
Determine the correct option Since the question asks for the correct answer, and we have two matching options (C and D), let's re-examine the question and the options. We have cot θ = 4 3 . Therefore, tan θ = c o t θ 1 = 4 3 1 = 3 4 . So option D is correct.
Option C is also correct, cos θ = − 5 3 . However, the question is likely expecting us to derive the value of cos θ from the given information, rather than directly using the given cot θ to find tan θ . But since both are correct, let's choose the one that is most directly derived from the given information and the quadrant.
Final Answer Both options C and D are correct. However, since the question implies there is only one correct answer, and option D is a direct consequence of the given information ( cot θ = 4 3 ), while option C requires calculating r and using the fact that we are in quadrant 3, we will choose option D.
State the final answer The correct answer is D. tan θ = 3 4 .
Examples
Understanding trigonometric functions and their relationships is crucial in fields like navigation and surveying. For instance, if a surveyor knows the cotangent of an angle of elevation and the quadrant in which the angle lies, they can determine the tangent of the angle to calculate heights of buildings or depths of valleys. This ensures accurate measurements and safe construction practices.
