If $-5 \sin \theta=4$ and $\theta \in(90,270)$, determine without the use of a calculator and with the aid of a diagram the value of:
(1) $5 \cos \theta-3 \tan \theta$
(2) $\frac{4}{\sin \theta}-\frac{3}{\cos \theta}$
Answer (1)
Determine sin θ from the given equation: sin θ = − 5 4 .
Find cos θ using the Pythagorean identity: cos θ = − 5 3 (since θ is in the third quadrant).
Calculate tan θ : tan θ = c o s θ s i n θ = 3 4 .
Evaluate the expressions: 5 cos θ − 3 tan θ = − 7 and s i n θ 4 − c o s θ 3 = 0 .
− 7 , 0
Explanation
Problem Analysis We are given that − 5 s in t h e t a = 4 and t h e t a in ( 9 0 c i rc , 27 0 c i rc ) . This means that t h e t a lies in either the second or third quadrant. We need to find the values of 5 cos t h e t a − 3 t an t h e t a and f r a c 4 s in t h e t a − f r a c 3 cos t h e t a without using a calculator.
Finding s in t h e t a First, let's find s in t h e t a . From the given equation, we have s in t h e t a = − f r a c 4 5 Since s in t h e t a is negative, t h e t a must lie in the third quadrant, i.e., 18 0 c i rc < t h e t a < 27 0 c i rc .
Finding cos t h e t a Now, we need to find cos t h e t a . We know that s i n 2 t h e t a + co s 2 t h e t a = 1 . Therefore, co s 2 t h e t a = 1 − s i n 2 t h e t a = 1 − l e f t ( − f r a c 4 5 r i g h t ) 2 = 1 − f r a c 16 25 = f r a c 9 25 Since t h e t a is in the third quadrant, cos t h e t a is negative. Thus, cos t h e t a = − s q r t f r a c 9 25 = − f r a c 3 5
Finding t an t h e t a Next, we find t an t h e t a . We know that t an t h e t a = f r a c s in t h e t a cos t h e t a . Therefore, t an t h e t a = f r a c − f r a c 4 5 − f r a c 3 5 = f r a c 4 3
Calculating 5 cos t h e t a − 3 t an t h e t a Now, we can find the value of the first expression, 5 cos t h e t a − 3 t an t h e t a :
5 cos t h e t a − 3 t an t h e t a = 5 l e f t ( − f r a c 3 5 r i g h t ) − 3 l e f t ( f r a c 4 3 r i g h t ) = − 3 − 4 = − 7
Calculating f r a c 4 s in t h e t a − f r a c 3 cos t h e t a Finally, we find the value of the second expression, f r a c 4 s in t h e t a − f r a c 3 cos t h e t a :
f r a c 4 s in t h e t a − f r a c 3 cos t h e t a = f r a c 4 − f r a c 4 5 − f r a c 3 − f r a c 3 5 = − 5 − ( − 5 ) = − 5 + 5 = 0
Final Answer Therefore, the values of the expressions are: (1) 5 cos t h e t a − 3 t an t h e t a = − 7 (2) f r a c 4 s in t h e t a − f r a c 3 cos t h e t a = 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, we use trigonometric functions to describe its angular displacement over time. Similarly, in electrical engineering, alternating current (AC) is modeled using sinusoidal functions. The ability to manipulate and evaluate trigonometric expressions without a calculator is a valuable skill in these contexts, allowing for quick estimations and problem-solving.
