Login
Premium
Home Question Bank Math Trigonometry
Question
upstudy study bank question image url

This questions 1 pomt(9) possible Find the exact value of each of the remaining trigonometric functions of 0 . Rationalize denominators when applicable, \( \tan \theta=-\frac{3}{4} \), given that 0 is in quadrant IV Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \sin \theta= \) \( \square \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \cos \theta= \) \( \square \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \csc 0= \) \( \square \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Ask by Carter Griffiths. in the United States
Mar 13,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

\[ \sin \theta = -\frac{3}{5}, \quad \cos \theta = \frac{4}{5}, \quad \csc \theta = -\frac{5}{3} \]

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Mind Expander

To find the trigonometric functions of \(\theta\) given that \(\tan \theta = -\frac{3}{4}\) and \(\theta\) is in quadrant IV, we start with the identity: \[ \tan \theta = \frac{\sin \theta}{\cos \theta}. \] Since \(\tan \theta = -\frac{3}{4}\), we can think of opposite side (y) as -3 and adjacent side (x) as 4. Using the Pythagorean theorem to find the hypotenuse \(r\): \[ r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \] Now we can find \(\sin \theta\) and \(\cos \theta\): \[ \sin \theta = \frac{y}{r} = \frac{-3}{5}, \quad \cos \theta = \frac{x}{r} = \frac{4}{5}. \] Now, rationalizing if needed: **Sin and Cos Values:** A. \(\sin \theta = -\frac{3}{5}\) A. \(\cos \theta = \frac{4}{5}\) To find \(\csc \theta\): \[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3}. \] So, A. \(\csc \theta = -\frac{5}{3}\)

Related Questions

Question2 Prove the following dentities a) \( \sin x / 1-\cos 2 x=1 / 2 \sin x \)
Trigonometry South Africa Mar 14, 2025
\[ \cos 50^{\circ} \cdot \tan (-x) \] 1. \( \cos ^{2}\left(180^{\circ}+x\right)+\cos (-x) \cdot \tan x \cdot \cos \left(90^{\circ}+x\right) \) j. \( \frac{\cos \left(90^{\circ}+x\right)}{\sin \left(x-180^{\circ}\right)+3 \sin (-x)} \) k. \( \sin \left(110^{\circ}-x\right) \cos \left(70^{\circ}-y\right)+\cos \left(110^{\circ}-x\right) \sin \left(70^{\circ}-y\right) \) ACTIVITY 3 (Groups) 1. Prove the identities and determine for which values is/are the identity undefined, where \( x \in\left[0^{\circ} ; 360^{\circ}\right] \) a) \( \frac{\sin 2 x-\cos 2 x+1}{\sin 2 x+\cos 2 x+1}=\tan x \) and hence evaluate \( \tan 22,5^{\circ} \) b) \( \frac{2 \sin ^{2} x+\cos x+1}{1-\cos \left(540^{\circ}+x\right)}=2 \cos x-1 \) c) \( \frac{\sin 2 x}{\cos 2 x+\sin 270^{\circ}}=-\frac{\cos x}{\sin x} \) d) \( \frac{\cos 2 x}{\sin x+\cos x}=\cos x-\sin x \) c) \( \frac{\sin x+\sin 2 x}{1+\cos x+\cos 2 x}=\tan x \) f) \( \quad(1-\tan A) \frac{\cos A}{\cos 2 A}=\frac{1}{\cos A+\sin A} \) B. \( \frac{\sin 2 x}{1+\cos 2 x} \cdot \frac{\sin 2 x}{1-\cos 2 x}=1 \) h. \( \frac{\sin 3 \theta}{\sin \theta}-\frac{\cos 3 \theta}{\cos \theta}=2 \) 1. \( \frac{1+\cos 2 x}{\cos 2 x}=\frac{\tan 2 x}{\tan x} \)
Trigonometry South Africa Mar 14, 2025
S. 2 Simplify the following expeasion to a single irigatominetric tein. \[ \frac{\sin \left(360^{2}-x\right) \cdot \tan (-x)}{\cos \left(1800^{3}+x\right)\left(\sin ^{2} A+\cos ^{2} A\right)} \] S.2. Simplify fully; \( \frac{\sin 192^{\circ} \cdot \tan 300^{\circ} \cos \left(2070^{2}+x\right)}{\sin \left(-x-100^{\circ}\right) \cdot \cos 107^{\circ}} \)
Trigonometry South Africa Mar 14, 2025
QUESTION 3 Applications 3.1 Express the following as single trigonometry ratio: 3.1.1 \( \cos 2 x \cdot \cos 3 x-\sin 2 x \cdot \sin 3 x \) 3.1.2 \( \sin 2 x \cdot \cos x+\cos 2 x \cdot \sin x \) \( \qquad \) 3.2 Determine the values of the following without using a calculator. (2) \( 3.2 .1 \sin 85^{\circ} \cdot \cos 25^{\circ}-\cos 85^{\circ} \cdot \sin 25^{\circ} \) \( \qquad \) \( \qquad \) \( 3.2 .2 \cos 160^{\circ} \cdot \cos 10^{\circ}+\sin 160^{\circ} \cdot \sin 10^{\circ} \) (4) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) [11] Grade 12 Mathematics SBA 2025 Page 15 of 29
Trigonometry South Africa Mar 14, 2025
b) \( \tan 405^{\circ}-\tan (\arctan 1)= \)
Trigonometry Slovenia Mar 14, 2025

Latest Trigonometry Questions

\[ \cos 50^{\circ} \cdot \tan (-x) \] 1. \( \cos ^{2}\left(180^{\circ}+x\right)+\cos (-x) \cdot \tan x \cdot \cos \left(90^{\circ}+x\right) \) j. \( \frac{\cos \left(90^{\circ}+x\right)}{\sin \left(x-180^{\circ}\right)+3 \sin (-x)} \) k. \( \sin \left(110^{\circ}-x\right) \cos \left(70^{\circ}-y\right)+\cos \left(110^{\circ}-x\right) \sin \left(70^{\circ}-y\right) \) ACTIVITY 3 (Groups) 1. Prove the identities and determine for which values is/are the identity undefined, where \( x \in\left[0^{\circ} ; 360^{\circ}\right] \) a) \( \frac{\sin 2 x-\cos 2 x+1}{\sin 2 x+\cos 2 x+1}=\tan x \) and hence evaluate \( \tan 22,5^{\circ} \) b) \( \frac{2 \sin ^{2} x+\cos x+1}{1-\cos \left(540^{\circ}+x\right)}=2 \cos x-1 \) c) \( \frac{\sin 2 x}{\cos 2 x+\sin 270^{\circ}}=-\frac{\cos x}{\sin x} \) d) \( \frac{\cos 2 x}{\sin x+\cos x}=\cos x-\sin x \) c) \( \frac{\sin x+\sin 2 x}{1+\cos x+\cos 2 x}=\tan x \) f) \( \quad(1-\tan A) \frac{\cos A}{\cos 2 A}=\frac{1}{\cos A+\sin A} \) B. \( \frac{\sin 2 x}{1+\cos 2 x} \cdot \frac{\sin 2 x}{1-\cos 2 x}=1 \) h. \( \frac{\sin 3 \theta}{\sin \theta}-\frac{\cos 3 \theta}{\cos \theta}=2 \) 1. \( \frac{1+\cos 2 x}{\cos 2 x}=\frac{\tan 2 x}{\tan x} \)
Trigonometry South Africa Mar 14, 2025
(2) \( \cos \left(\theta+10^{\circ}\right)=-\cos \theta ; \quad \theta \in\left(-180^{\circ} ; 180^{\circ}\right) \)
Trigonometry South Africa Mar 14, 2025
Dokaži: \( \quad 1-\frac{\sin ^{2}(2 x)}{2 \cos ^{2} x}=\sin \left(\frac{\pi}{2}-2 x\right) \)
Trigonometry Slovenia Mar 14, 2025
S. 2 Simplify the following expeasion to a single irigatominetric tein. \[ \frac{\sin \left(360^{2}-x\right) \cdot \tan (-x)}{\cos \left(1800^{3}+x\right)\left(\sin ^{2} A+\cos ^{2} A\right)} \] S.2. Simplify fully; \( \frac{\sin 192^{\circ} \cdot \tan 300^{\circ} \cos \left(2070^{2}+x\right)}{\sin \left(-x-100^{\circ}\right) \cdot \cos 107^{\circ}} \)
Trigonometry South Africa Mar 14, 2025
\( \tan \theta=3 \) and \( \cos \theta<0 \), use a sketch in the correct quadrant to determine the we of: \( \frac{\sin \theta+\cos \theta}{2 \sin \theta} \) without using a calculator.
Trigonometry South Africa Mar 14, 2025
Prev Next
Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy