Studdy AI Logo
Math  /  Trigonometry

Questionb. tan⁡xcos⁡xsin⁡x−1=0\tan x \cos x \sin x-1=0 where 0∘≤x≤360∘0^{\circ} \leq x \leq 360^{\circ}

Studdy Solution

STEP 1

1. The equation involves trigonometric functions: tangent, cosine, and sine.
2. We will use trigonometric identities to simplify and solve the equation.
3. The solution will be found within the interval 0∘≤x≤360∘0^{\circ} \leq x \leq 360^{\circ}.

STEP 2

1. Use trigonometric identities to simplify the equation.
2. Solve the simplified equation for x x .
3. Verify solutions within the given interval.

STEP 3

Start by using the identity tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x} to rewrite the equation:
tan⁡xcos⁡xsin⁡x−1=0\tan x \cos x \sin x - 1 = 0
Substitute tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}:
sin⁡xcos⁡xcos⁡xsin⁡x−1=0\frac{\sin x}{\cos x} \cos x \sin x - 1 = 0
Simplify the expression:
sin⁡2x−1=0\sin^2 x - 1 = 0

STEP 4

Solve the equation sin⁡2x−1=0\sin^2 x - 1 = 0:
sin⁡2x=1\sin^2 x = 1
Take the square root of both sides:
sin⁡x=±1\sin x = \pm 1

STEP 5

Determine the angles x x where sin⁡x=1\sin x = 1 and sin⁡x=−1\sin x = -1 within the interval 0∘≤x≤360∘0^{\circ} \leq x \leq 360^{\circ}:
- sin⁡x=1\sin x = 1 at x=90∘x = 90^{\circ} - sin⁡x=−1\sin x = -1 at x=270∘x = 270^{\circ}
Thus, the solutions are:
x=90∘,270∘ x = 90^{\circ}, 270^{\circ}
The values of x x are:
90∘,270∘ \boxed{90^{\circ}, 270^{\circ}}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord