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Activity 2 Simplify the following WITHOUT the use of a calculator: \( 2.1 \frac{\tan \left(180^{\circ}+x\right) \cdot \cos \left(360^{\circ}-x\right)}{\sin (180-x) \cdot \cos \left(90^{\circ}+x\right)+\cos \left(540^{\circ}+x\right) \cdot \cos (-x)} \) \( 2.3 \sin \left(90^{\circ}-x\right) \cdot \cos \left(180^{\circ}-x\right)+\tan x \cdot \cos (-x) \cdot \sin \left(180^{\circ}+x\right) \) \( 2.4 \frac{\sin ^{2} \theta}{\sin \left(180^{\circ}-\theta\right) \cdot \cos \left(90^{\circ}+\theta\right)+\tan 45^{\circ}} \) \( 2.5 \frac{\sin 104^{\circ}\left(2 \cos ^{2} 15^{\circ}\right.}{\tan 38^{\circ} \cdot \sin ^{2} 412^{\circ}} \) \( 2.6 \frac{\sin \left(90^{\circ}+\theta\right)+\cos \left(180^{\circ}+\theta\right) \sin (-\theta)}{\sin 180^{\circ}-\tan 100^{\circ}} \)

Ask by Griffiths Gibson. in South Africa
Feb 27,2025

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Answer

Here are the simplified expressions: 1. **2.1:** \(\frac{\sin(x)}{-\sin^2(x) + \cos^2(x)}\) 2. **2.3:** \(-1\) 3. **2.4:** \(\frac{\sin^2 \theta}{-\sin^2 \theta + 1}\) 4. **2.5:** \(\frac{\sin(104^{\circ}) \cdot (2 \cos^2(15^{\circ}))}{\tan(38^{\circ}) \cdot \sin^2(52^{\circ})}\) 5. **2.6:** \(\frac{\cos(\theta) - \cos(\theta) \sin(\theta)}{-\tan(100^{\circ})}\)

Solution

Simplify the expression by following steps: - step0: Solution: \(\frac{2.6\left(\sin\left(90+\theta \right)+\cos\left(180+\theta \right)\sin\left(-\theta \right)\right)}{\left(\sin\left(180\right)-\tan\left(100\right)\right)}\) - step1: Remove the parentheses: \(\frac{2.6\left(\sin\left(90+\theta \right)+\cos\left(180+\theta \right)\sin\left(-\theta \right)\right)}{\sin\left(180\right)-\tan\left(100\right)}\) - step2: Rewrite the expression: \(\frac{2.6\left(\sin\left(90+\theta \right)+\cos\left(\theta +180\right)\sin\left(-\theta \right)\right)}{\sin\left(180\right)-\tan\left(100\right)}\) - step3: Transform the expression: \(\frac{2.6\left(\sin\left(90+\theta \right)+\cos\left(\theta +180\right)\left(-\sin\left(\theta \right)\right)\right)}{\sin\left(180\right)-\tan\left(100\right)}\) - step4: Rewrite the expression: \(\frac{2.6\left(\sin\left(\theta +90\right)+\cos\left(\theta +180\right)\left(-\sin\left(\theta \right)\right)\right)}{\sin\left(180\right)-\tan\left(100\right)}\) - step5: Rewrite the expression: \(\frac{2.6\left(\sin\left(\theta +90\right)-\cos\left(\theta +180\right)\sin\left(\theta \right)\right)}{\sin\left(180\right)-\tan\left(100\right)}\) - step6: Multiply the terms: \(\frac{2.6\sin\left(\theta +90\right)-2.6\cos\left(\theta +180\right)\sin\left(\theta \right)}{\sin\left(180\right)-\tan\left(100\right)}\) - step7: Convert the expressions: \(\frac{\frac{13}{5}\sin\left(\theta +90\right)-\frac{13}{5}\cos\left(\theta +180\right)\sin\left(\theta \right)}{\sin\left(180\right)-\tan\left(100\right)}\) - step8: Simplify the expression: \(\frac{13\sin\left(\theta +90\right)-13\cos\left(\theta +180\right)\sin\left(\theta \right)}{5\sin\left(180\right)-5\tan\left(100\right)}\) - step9: Rewrite the expression: \(\left(13\sin\left(\theta +90\right)-13\cos\left(\theta +180\right)\sin\left(\theta \right)\right)\left(5\sin\left(180\right)-5\tan\left(100\right)\right)^{-1}\) - step10: Calculate: \(\left(13\sin\left(\theta +90\right)-13\cos\left(\theta +180\right)\sin\left(\theta \right)\right)\times \frac{1}{5\sin\left(180\right)-5\tan\left(100\right)}\) - step11: Calculate: \(\frac{13}{5\sin\left(180\right)-5\tan\left(100\right)}\times \sin\left(\theta +90\right)+\frac{13}{5\tan\left(100\right)-5\sin\left(180\right)}\times \cos\left(\theta +180\right)\sin\left(\theta \right)\) - step12: Rewrite the expression: \(\frac{13}{5\sin\left(180\right)-5\tan\left(100\right)}\times \sin\left(\theta +90\right)+\frac{13}{-5\sin\left(180\right)+5\tan\left(100\right)}\times \sin\left(\theta \right)\cos\left(\theta +180\right)\) - step13: Simplify: \(\frac{13\sin\left(\theta +90\right)-13\sin\left(\theta \right)\cos\left(\theta +180\right)}{5\sin\left(180\right)-5\tan\left(100\right)}\) Calculate or simplify the expression \( 2.5 * (\sin(104) * (2 * \cos^2(15))) / (\tan(38) * \sin^2(412)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{2.5\left(\sin\left(104\right)\left(2\cos^{2}\left(15\right)\right)\right)}{\left(\tan\left(38\right)\sin^{2}\left(412\right)\right)}\) - step1: Remove the parentheses: \(\frac{2.5\sin\left(104\right)\times 2\cos^{2}\left(15\right)}{\tan\left(38\right)\sin^{2}\left(412\right)}\) - step2: Multiply the terms: \(\frac{5\sin\left(104\right)\cos^{2}\left(15\right)}{\tan\left(38\right)\sin^{2}\left(412\right)}\) - step3: Calculate the trigonometric value: \(-15.719411\) Let's simplify each expression step by step without using a calculator. ### 2.1 \[ \frac{\tan(180^{\circ}+x) \cdot \cos(360^{\circ}-x)}{\sin(180-x) \cdot \cos(90^{\circ}+x) + \cos(540^{\circ}+x) \cdot \cos(-x)} \] **Step 1: Simplify the trigonometric functions.** - \(\tan(180^{\circ}+x) = \tan(x)\) - \(\cos(360^{\circ}-x) = \cos(x)\) - \(\sin(180-x) = \sin(x)\) - \(\cos(90^{\circ}+x) = -\sin(x)\) - \(\cos(540^{\circ}+x) = \cos(x)\) - \(\cos(-x) = \cos(x)\) **Step 2: Substitute these values into the expression.** \[ \frac{\tan(x) \cdot \cos(x)}{\sin(x) \cdot (-\sin(x)) + \cos(x) \cdot \cos(x)} \] This simplifies to: \[ \frac{\tan(x) \cdot \cos(x)}{-\sin^2(x) + \cos^2(x)} \] **Step 3: Use the identity \(\tan(x) = \frac{\sin(x)}{\cos(x)}\)** \[ \frac{\frac{\sin(x)}{\cos(x)} \cdot \cos(x)}{-\sin^2(x) + \cos^2(x)} = \frac{\sin(x)}{-\sin^2(x) + \cos^2(x)} \] ### 2.3 \[ \sin(90^{\circ}-x) \cdot \cos(180^{\circ}-x) + \tan(x) \cdot \cos(-x) \cdot \sin(180^{\circ}+x) \] **Step 1: Simplify the trigonometric functions.** - \(\sin(90^{\circ}-x) = \cos(x)\) - \(\cos(180^{\circ}-x) = -\cos(x)\) - \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) - \(\cos(-x) = \cos(x)\) - \(\sin(180^{\circ}+x) = -\sin(x)\) **Step 2: Substitute these values into the expression.** \[ \cos(x) \cdot (-\cos(x)) + \frac{\sin(x)}{\cos(x)} \cdot \cos(x) \cdot (-\sin(x)) \] This simplifies to: \[ -\cos^2(x) - \sin^2(x) = -1 \] ### 2.4 \[ \frac{\sin^2 \theta}{\sin(180^{\circ}-\theta) \cdot \cos(90^{\circ}+\theta) + \tan 45^{\circ}} \] **Step 1: Simplify the trigonometric functions.** - \(\sin(180^{\circ}-\theta) = \sin(\theta)\) - \(\cos(90^{\circ}+\theta) = -\sin(\theta)\) - \(\tan(45^{\circ}) = 1\) **Step 2: Substitute these values into the expression.** \[ \frac{\sin^2 \theta}{\sin(\theta) \cdot (-\sin(\theta)) + 1} = \frac{\sin^2 \theta}{-\sin^2 \theta + 1} \] ### 2.5 \[ \frac{\sin(104^{\circ}) \cdot (2 \cos^2(15^{\circ}))}{\tan(38^{\circ}) \cdot \sin^2(412^{\circ})} \] **Step 1: Simplify the trigonometric functions.** - \(\sin(412^{\circ}) = \sin(52^{\circ})\) (since \(412 - 360 = 52\)) **Step 2: Substitute these values into the expression.** \[ \frac{\sin(104^{\circ}) \cdot (2 \cos^2(15^{\circ}))}{\tan(38^{\circ}) \cdot \sin^2(52^{\circ})} \] ### 2.6 \[ \frac{\sin(90^{\circ}+\theta) + \cos(180^{\circ}+\theta) \sin(-\theta)}{\sin(180^{\circ}) - \tan(100^{\circ})} \] **Step 1: Simplify the trigonometric functions.** - \(\sin(90^{\circ}+\theta) = \cos(\theta)\) - \(\cos(180^{\circ}+\theta) = -\cos(\theta)\) - \(\sin(-\theta) = -\sin(\theta)\) - \(\sin(180^{\circ}) = 0\) **Step 2: Substitute these values into the expression.** \[ \frac{\cos(\theta) - \cos(\theta) \sin(\theta)}{0 - \tan(100^{\circ})} \] ### Summary of Results - **2.1:** \(\frac{\sin(x)}{-\sin^2(x) + \cos^2(x)}\) - **2.3:** \(-1\) - **2.4:** \(\frac{\sin^2 \theta}{-\sin^2 \theta + 1}\) - **2.5:** \(\frac{\sin(104^{\circ}) \cdot (2 \cos^2(15^{\circ}))}{\tan(38^{\circ}) \cdot \sin^2(52^{\circ})}\) - **2.6:** \(\frac{\cos(\theta) - \cos(\theta) \sin(\theta)}{-\tan(100^{\circ})}\)

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Beyond the Answer

Did you know that trigonometric identities can significantly simplify complex problems? These relationships, like \( \sin(90^\circ - x) = \cos(x) \) and \( \tan(180^\circ + x) = \tan(x) \), help us manipulate and reduce expressions efficiently. Understanding and applying these can save you tons of time when solving problems, turning math into more of a puzzle than a chore! Another fun nugget is the historical journey of trigonometry. It dates back to ancient civilizations like the Babylonians and Greeks, who utilized these concepts in various ways—like astronomy and architecture! Famous figures such as Hipparchus and Ptolemy made astronomical calculations involving trigonometric functions that laid the foundation for modern mathematics. So, every time you tackle a trigonometry problem, you’re participating in a centuries-old tradition!

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