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1. Given: (i) \( A=60^{\circ} \) and \( B=30^{\circ} \) (ii) \( A=110^{\circ} \) and \( B=50^{\circ} \) (iii) \( A=225^{\circ} \) and \( B=135^{\circ} \) Use a calculator to evaluate each of the following: (a) \( \cos (A-B) \) (b) \( \cos A-\cos B \) (c) \( \cos A \cdot \cos B+\sin A \cdot \sin B \) .2 Compare the values of \( \cos (A-B) \) with the values of \( \cos A-\cos B \)

Ask by Elliott Hill. in South Africa
Feb 27,2025

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Answer

For each set of angles, calculate \( \cos(A - B) \), \( \cos A - \cos B \), and \( \cos A \cdot \cos B + \sin A \cdot \sin B \). The results show that \( \cos(A - B) \) and \( \cos A \cdot \cos B + \sin A \cdot \sin B \) are often similar, while \( \cos A - \cos B \) can differ significantly.

Solution

Calculate the value by following steps: - step0: Calculate: \(\cos\left(60-30\right)\) - step1: Subtract the numbers: \(\cos\left(30\right)\) Calculate or simplify the expression \( \cos(60)*\cos(30)+\sin(60)*\sin(30) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(60\right)\cos\left(30\right)+\sin\left(60\right)\sin\left(30\right)\) - step1: Transform the expression: \(\cos\left(60-30\right)\) - step2: Subtract the numbers: \(\cos\left(30\right)\) Calculate or simplify the expression \( \cos(225)*\cos(135)+\sin(225)*\sin(135) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(225\right)\cos\left(135\right)+\sin\left(225\right)\sin\left(135\right)\) - step1: Transform the expression: \(\cos\left(225-135\right)\) - step2: Subtract the numbers: \(\cos\left(90\right)\) Calculate or simplify the expression \( \cos(110-50) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(110-50\right)\) - step1: Subtract the numbers: \(\cos\left(60\right)\) Calculate or simplify the expression \( \cos(110)*\cos(50)+\sin(110)*\sin(50) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(110\right)\cos\left(50\right)+\sin\left(110\right)\sin\left(50\right)\) - step1: Transform the expression: \(\cos\left(110-50\right)\) - step2: Subtract the numbers: \(\cos\left(60\right)\) Calculate or simplify the expression \( \cos(225-135) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(225-135\right)\) - step1: Subtract the numbers: \(\cos\left(90\right)\) Calculate or simplify the expression \( \cos(225)-\cos(135) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(225\right)-\cos\left(135\right)\) - step1: Calculate the trigonometric value: \(1.363407\) Calculate or simplify the expression \( \cos(110)-\cos(50) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(110\right)-\cos\left(50\right)\) - step1: Calculate the trigonometric value: \(-1.963987\) Calculate or simplify the expression \( \cos(60)-\cos(30) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(60\right)-\cos\left(30\right)\) - step1: Calculate the trigonometric value: \(-1.106664\) Let's evaluate the expressions step by step for each set of angles \( A \) and \( B \). ### For \( A = 60^{\circ} \) and \( B = 30^{\circ} \): 1. **Calculate \( \cos(A - B) \)**: \[ \cos(60 - 30) = \cos(30) \approx 0.154251 \] 2. **Calculate \( \cos A - \cos B \)**: \[ \cos(60) - \cos(30) \approx -1.106664 \] 3. **Calculate \( \cos A \cdot \cos B + \sin A \cdot \sin B \)**: \[ \cos(60) \cdot \cos(30) + \sin(60) \cdot \sin(30) \approx 0.154251 \] ### For \( A = 110^{\circ} \) and \( B = 50^{\circ} \): 1. **Calculate \( \cos(A - B) \)**: \[ \cos(110 - 50) = \cos(60) \approx -0.952413 \] 2. **Calculate \( \cos A - \cos B \)**: \[ \cos(110) - \cos(50) \approx -1.963987 \] 3. **Calculate \( \cos A \cdot \cos B + \sin A \cdot \sin B \)**: \[ \cos(110) \cdot \cos(50) + \sin(110) \cdot \sin(50) \approx -0.952413 \] ### For \( A = 225^{\circ} \) and \( B = 135^{\circ} \): 1. **Calculate \( \cos(A - B) \)**: \[ \cos(225 - 135) = \cos(90) \approx 0 \] 2. **Calculate \( \cos A - \cos B \)**: \[ \cos(225) - \cos(135) \approx 1.363407 \] 3. **Calculate \( \cos A \cdot \cos B + \sin A \cdot \sin B \)**: \[ \cos(225) \cdot \cos(135) + \sin(225) \cdot \sin(135) \approx 0 \] ### Summary of Results: - For \( A = 60^{\circ} \) and \( B = 30^{\circ} \): - \( \cos(A - B) \approx 0.154251 \) - \( \cos A - \cos B \approx -1.106664 \) - \( \cos A \cdot \cos B + \sin A \cdot \sin B \approx 0.154251 \) - For \( A = 110^{\circ} \) and \( B = 50^{\circ} \): - \( \cos(A - B) \approx -0.952413 \) - \( \cos A - \cos B \approx -1.963987 \) - \( \cos A \cdot \cos B + \sin A \cdot \sin B \approx -0.952413 \) - For \( A = 225^{\circ} \) and \( B = 135^{\circ} \): - \( \cos(A - B) \approx 0 \) - \( \cos A - \cos B \approx 1.363407 \) - \( \cos A \cdot \cos B + \sin A \cdot \sin B \approx 0 \) ### Comparison: - In the first two cases, \( \cos(A - B) \) and \( \cos A \cdot \cos B + \sin A \cdot \sin B \) yield similar results, while \( \cos A - \cos B \) diverges significantly. - In the third case, \( \cos(A - B) \) and \( \cos A \cdot \cos B + \sin A \cdot \sin B \) are both zero, while \( \cos A - \cos B \) is positive. This shows that while \( \cos(A - B) \) and \( \cos A \cdot \cos B + \sin A \cdot \sin B \) are often related, \( \cos A - \cos B \) can differ significantly.

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The Deep Dive

For each pair of angles, let’s dive into the calculations! (i) For \( A=60^{\circ} \) and \( B=30^{\circ} \): - (a) \( \cos(60^\circ - 30^\circ) = \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866 \) - (b) \( \cos(60^\circ) - \cos(30^\circ) = \frac{1}{2} - \frac{\sqrt{3}}{2} \approx -0.366 \) - (c) \( \cos(60^\circ) \cdot \cos(30^\circ) + \sin(60^\circ) \cdot \sin(30^\circ) = \frac{1}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{1}{2} = \frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \approx 0.866 \) (ii) For \( A=110^{\circ} \) and \( B=50^{\circ} \): - (a) \( \cos(110^\circ - 50^\circ) = \cos(60^\circ) = \frac{1}{2} \) - (b) \( \cos(110^\circ) - \cos(50^\circ) \approx -0.342 - 0.643 \approx -0.985 \) - (c) \( \cos(110^\circ)\cdot\cos(50^\circ) + \sin(110^\circ)\cdot\sin(50^\circ \approx -0.342\cdot0.643 + 0.939\cdot0.766 \approx 0.526 \) (iii) For \( A=225^{\circ} \) and \( B=135^{\circ} \): - (a) \( \cos(225^\circ - 135^\circ) = \cos(90^\circ) = 0 \) - (b) \( \cos(225^\circ) - \cos(135^\circ) \approx -0.707 - (-0.707) = 0 \) - (c) \( \cos(225^\circ)\cdot\cos(135^\circ) + \sin(225^\circ)\cdot\sin(135^\circ) \approx (-0.707)(-0.707) + (-0.707)(0.707) = 0.5 - 0.5 = 0 \) In all cases, \( \cos(A-B) \) and \( \cos A - \cos B \) have some fascinating relationships! While they sometimes yield similar results, the cosine difference formula brings distinct outcomes depending on angle values. Always double-check those calculations, and remember, using a calculator helps avoid common mistakes!

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