Given: (i) \( A=60^{\circ} \) and \( B=30^{\circ} \) (ii) \( A=110^{\circ} \) and \( B=50^{\circ} \) (iti) \( A=225^{\circ} \) and \( B=135^{\circ} \) Use a calculator to evaluate each of the following (a) \( \cos (A-B) \) (b) \( \cos A-\cos B \) (c) \( \cos A \cos B+\sin A \sin B \)
Real Tutor Solution
Answer
Solution
\( -0.3420 - 0.6428 = -0.9848 \) | \( (-0.3420)(0.6428) + (0.9397)(0.7660) \approx -0.2195 + 0.7203 = 0.5008 \approx 0.5000 \) | | **(iii) \( A = 225^{\circ} \), \( B = 135^{\circ} \)** | | Calculation | \( \cos(225^{\circ} - 135^{\circ}) = \cos(90^{\circ}) \) | \( \cos(225^{\circ}) - \cos(135^{\circ}) \) | \( \cos(225^{\circ}) \cos(135^{\circ}) + \sin(225^{\circ}) \sin(135^{\circ}) \) | | Value | \( \cos(90^{\circ}) = 0.0000 \) | \( \cos(225^{\circ}) \approx -0.7071 \), \( \cos(135^{\circ}) \approx -0.7071 \)
\( -0.7071 - (-0.7071) = 0.0000 \) | \( (-0.7071)(-0.7071) + (-0.7071)(0.7071) = 0.5000 - 0.5000 = 0.0000 \) | ### Summary of Results: | **Case** | **(a) \( \cos(A - B) \)** | **(b) \( \cos A - \cos B \)** | **(c) \( \cos A \cos B + \sin A \sin B \)** | |----------|----------------------------|---------------------------------|-----------------------------------------------| | **(i)** | 0.8660 | -0.3660 | 0.8660 | | **(ii)** | 0.5000 | -0.9848 | 0.5000 | | **(iii)** | 0.0000 | 0.0000 | 0.0000 | ### Observations: - **Part (a)** and **Part (c)** yield the same results for each case. This is consistent with the trigonometric identity: \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \] - **Part (b)** provides a different relationship, calculating the difference between the cosines of angles \( A \) and \( B \). Feel free to reach out if you need further explanations or additional assistance!
Reviewed and approved by the UpStudy tutoring team
Explain 
Simplify this solution Mind Expander
To evaluate the expressions with the given angles, let's work through each pair of angles step by step: ### For \( A = 60^{\circ} \) and \( B = 30^{\circ} \): 1. **(a)** \( \cos (A-B) = \cos(60^{\circ} - 30^{\circ}) = \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \approx 0.866 \) 2. **(b)** \( \cos A - \cos B = \cos(60^{\circ}) - \cos(30^{\circ}) = \frac{1}{2} - \frac{\sqrt{3}}{2} \approx -0.366 \) 3. **(c)** \( \cos A \cos B + \sin A \sin B = \cos(60^{\circ}) \cos(30^{\circ}) + \sin(60^{\circ}) \sin(30^{\circ}) \) = \( \frac{1}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{1}{2} = \frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \approx 0.866 \) ### For \( A = 110^{\circ} \) and \( B = 50^{\circ} \): 1. **(a)** \( \cos (A-B) = \cos(110^{\circ} - 50^{\circ}) = \cos(60^{\circ}) = \frac{1}{2} = 0.5 \) 2. **(b)** \( \cos A - \cos B = \cos(110^{\circ}) - \cos(50^{\circ}) \approx -0.342 - 0.643 \approx -0.985 \) 3. **(c)** \( \cos A \cos B + \sin A \sin B \) = \( \cos(110^{\circ}) \cos(50^{\circ}) + \sin(110^{\circ}) \sin(50^{\circ}) \) = \( (-0.342)(0.643) + (0.940)(0.766) \approx -0.220 + 0.719 = 0.499 \approx 0.5 \) ### For \( A = 225^{\circ} \) and \( B = 135^{\circ} \): 1. **(a)** \( \cos (A-B) = \cos(225^{\circ} - 135^{\circ}) = \cos(90^{\circ}) = 0 \) 2. **(b)** \( \cos A - \cos B = \cos(225^{\circ}) - \cos(135^{\circ}) = -\frac{\sqrt{2}}{2} - (-\frac{\sqrt{2}}{2}) = 0 \) 3. **(c)** \( \cos A \cos B + \sin A \sin B \) = \( \cos(225^{\circ}) \cos(135^{\circ}) + \sin(225^{\circ}) \sin(135^{\circ}) \) = \( (-\frac{\sqrt{2}}{2})(-\frac{\sqrt{2}}{2}) + (-\frac{\sqrt{2}}{2})(\frac{\sqrt{2}}{2}) = \frac{1}{2} - \frac{1}{2} = 0 \) ### Summary of Results: - For \( A = 60^\circ \), \( B = 30^\circ \): - (a) \( \frac{\sqrt{3}}{2} \approx 0.866 \) - (b) \( \frac{1}{2} - \frac{\sqrt{3}}{2} \approx -0.366 \) - (c) \( \frac{\sqrt{3}}{2} \approx 0.866 \) - For \( A = 110^\circ \), \( B = 50^\circ \): - (a) \( 0.5 \) - (b) \( -0.985 \) - (c) \( 0.5 \) - For \( A = 225^\circ \), \( B = 135^\circ \): - (a) \( 0 \) - (b) \( 0 \)
