If [tex]$-5 \sin \theta=4$[/tex] and [tex]$\theta \in (90,270)$[/tex], determine, without the use of a calculator and with the aid of a diagram, the value of
(1) [tex]$5 \cos \theta-3 \tan \theta$[/tex]
(2) [tex]$\frac{4}{\sin \theta}-\frac{3}{\cos \theta}$[/tex]
Answer (1)
Find s in ( θ ) from the given equation: s in ( θ ) = − 5 4 .
Find cos ( θ ) using the Pythagorean identity and the quadrant information: cos ( θ ) = − 5 3 .
Calculate t an ( θ ) using s in ( θ ) and cos ( θ ) : t an ( θ ) = 3 4 .
Evaluate the expressions: 5 cos ( θ ) − 3 t an ( θ ) = − 7 and s in ( θ ) 4 − cos ( θ ) 3 = 0 .
The final answers are − 7 and 0 .
Explanation
Problem Setup We are given that − 5"." s in ( θ ) = 4 and θ ∈ ( 90 , 270 ) . We need to find the values of 5 cos ( θ ) − 3 t an ( θ ) and s in ( θ ) 4 − cos ( θ ) 3 .
Finding Sine First, we find s in ( θ ) from the given equation: s in ( θ ) = − 5 4 . Since θ is in the interval ( 90 , 270 ) , it lies either in the second or third quadrant. Since s in ( θ ) is negative, θ must be in the third quadrant ( 180 , 270 ) .
Finding Cosine Next, we find cos ( θ ) using the Pythagorean identity s i n 2 ( θ ) + co s 2 ( θ ) = 1 . Since θ is in the third quadrant, cos ( θ ) is also negative. Thus, cos ( θ ) = − 1 − s i n 2 ( θ ) = − 1 − ( − 5 4 ) 2 = − 1 − 25 16 = − 25 9 = − 5 3 .
Finding Tangent Now, we find t an ( θ ) using the identity t an ( θ ) = cos ( θ ) s in ( θ ) . t an ( θ ) = − 5 3 − 5 4 = 3 4 .
Evaluating Expression 1 Now we can evaluate the first expression: 5 cos ( θ ) − 3 t an ( θ ) = 5 ( − 5 3 ) − 3 ( 3 4 ) = − 3 − 4 = − 7 .
Evaluating Expression 2 Next, we evaluate the second expression: s in ( θ ) 4 − cos ( θ ) 3 = − 5 4 4 − − 5 3 3 = − 5 − ( − 5 ) = − 5 + 5 = 0 .
Final Answer Therefore, the values of the expressions are: (1) 5 cos ( θ ) − 3 t an ( θ ) = − 7 (2) s in ( θ ) 4 − cos ( θ ) 3 = 0
Examples
Understanding trigonometric functions and their relationships is crucial in various fields such as physics, engineering, and navigation. For instance, when analyzing the motion of a pendulum or the trajectory of a projectile, trigonometric functions help describe the angles and distances involved. Similarly, in electrical engineering, these functions are used to model alternating current (AC) circuits. By mastering these concepts, students can apply them to solve real-world problems involving oscillations, waves, and periodic phenomena.
