Let [tex]\cos x=\frac{1}{3}[/tex]. Find all possible values of [tex]\frac{\sec x-\tan x}{\sin x}[/tex]
Answer (2)
Rewrite the expression s i n x s e c x − t a n x in terms of sin x and cos x as c o s x s i n x 1 − s i n x .
Find sin x using the Pythagorean identity sin 2 x + cos 2 x = 1 , given cos x = 3 1 , which gives sin x = ± 3 2 2 .
Substitute both values of sin x into the expression c o s x s i n x 1 − s i n x to find the two possible values.
The possible values are 4 9 2 − 3 and − 4 9 2 − 3 , so the final answer is 4 9 2 − 3 , − 4 9 2 − 3 .
Explanation
Problem Analysis We are given that cos x = 3 1 and we need to find all possible values of the expression s i n x s e c x − t a n x .
Rewriting the Expression First, let's rewrite the expression in terms of sin x and cos x . Recall that sec x = c o s x 1 and tan x = c o s x s i n x . Therefore, sin x sec x − tan x = sin x c o s x 1 − c o s x s i n x = sin x c o s x 1 − s i n x = cos x sin x 1 − sin x .
Finding sin x Since cos x = 3 1 , we can find sin x using the Pythagorean identity sin 2 x + cos 2 x = 1 . Thus, sin 2 x = 1 − cos 2 x = 1 − ( 3 1 ) 2 = 1 − 9 1 = 9 8 . Taking the square root of both sides, we get sin x = ± 9 8 = ± 3 2 2 .
Case 1: Positive sin x Now we have two possible values for sin x : sin x = 3 2 2 and sin x = − 3 2 2 . Let's substitute these values into the expression c o s x s i n x 1 − s i n x along with cos x = 3 1 .
Case 1: sin x = 3 2 2 \frac{1 - \frac{2\sqrt{2}}{3}}{\frac{1}{3} \cdot \frac{2\sqrt{2}}{3}} = \frac{1 - \frac{2\sqrt{2}}{3}}{\frac{2\sqrt{2}}{9}} = \frac{\frac{3 - 2\sqrt{2}}{3}}{\frac{2\sqrt{2}}{9}} = \frac{3 - 2\sqrt{2}}{3} \cdot \frac{9}{2\sqrt{2}} = \frac{3(3 - 2\sqrt{2})}{2\sqrt{2}} = \frac{9 - 6\sqrt{2}}{2\sqrt{2}} = \frac{(9 - 6\sqrt{2})\sqrt{2}}{2\sqrt{2}\sqrt{2}} = \frac{9\sqrt{2} - 12}{4} = \frac{9\sqrt{2}}{4} - 3 \approx 0.182.$ 5. Case 2: Negative sin x Case 2: $\sin x = -\frac{2\sqrt{2}}{3}$ \frac{1 - \left(-\frac{2\sqrt{2}}{3}\right)}{\frac{1}{3} \cdot \left(-\frac{2\sqrt{2}}{3}\right)} = \frac{1 + \frac{2\sqrt{2}}{3}}{-\frac{2\sqrt{2}}{9}} = \frac{\frac{3 + 2\sqrt{2}}{3}}{-\frac{2\sqrt{2}}{9}} = \frac{3 + 2\sqrt{2}}{3} \cdot \frac{-9}{2\sqrt{2}} = \frac{-3(3 + 2\sqrt{2})}{2\sqrt{2}} = \frac{-9 - 6\sqrt{2}}{2\sqrt{2}} = \frac{(-9 - 6\sqrt{2})\sqrt{2}}{2\sqrt{2}\sqrt{2}} = \frac{-9\sqrt{2} - 12}{4} = -\frac{9\sqrt{2}}{4} - 3 \approx -6.182.$
Final Answer Therefore, the possible values of the expression are 4 9 2 − 3 and − 4 9 2 − 3 .
Examples
Understanding trigonometric identities and how to manipulate them is crucial in many fields, such as physics and engineering. For example, when analyzing the motion of a pendulum or the behavior of alternating current in an electrical circuit, trigonometric functions and their relationships are essential tools. By simplifying complex expressions involving trigonometric functions, engineers and physicists can more easily model and predict the behavior of these systems.
The possible values of the expression s i n x s e c x − t a n x given that cos x = 3 1 are 4 9 2 − 3 and − 4 9 2 − 3 .
;
