Grade 11 General Trigonometry questions, source [Mind Action Series Mathematics Grade 11 New Edition]
1 Answer
Lets write
\[p=(\sin \theta +\cos \theta)^2=\sin^2 \theta +\cos^2 \theta +2\sin \theta \cos \theta=1+2\sin \theta \cos \theta.\]
So
\[2\sin \theta \cos \theta=p-1. (1)\]
1. We have
\[(\sin \theta -\cos \theta)^2=\sin^2 \theta +\cos^2 \theta +2\sin \theta \cos \theta\]
\[=1-2\sin \theta \cos \theta=1-(p-1)=2-p.\]
Thus
\[\sin \theta -\cos \theta = \pm \sqrt{2-p}.\]
2. We have
\[(\sin \theta +\cos \theta)^3=\sin^3 \theta +\cos^3\theta+ 3\sin^2 \theta \cos \theta+3\sin\theta \cos^2 \theta\]
\[=\sin^3 \theta +\cos^3\theta+ 3\sin \theta \cos \theta(\sin \theta +\cos \theta ).\]
Hence
\[\sin^3 \theta +\cos^3\theta =(\sin \theta +\cos \theta)^3-3\sin \theta \cos \theta(\sin \theta +\cos \theta )\]
\[=(\sin \theta +\cos \theta) [(\sin \theta +\cos \theta)^2-3\sin \theta \cos \theta]\]
\[=(\sin \theta +\cos \theta) [(\sin \theta +\cos \theta)^2-\frac{3}{2} (2\sin \theta \cos \theta)]\]
\[=\pm \sqrt{p}(p-\frac{3}{2}(p-1))=\pm \sqrt{p}(-\frac{1}{2}p+\frac{3}{2})).\]
Thus
\[\sin^3 \theta +\cos^3\theta=\pm \sqrt{p}(-\frac{1}{2}p+\frac{3}{2}).\]
Erdos
- 1 Answer
- 514 views
- Pro Bono
Related Questions
- Need formula help
- Evaluate $\int_0^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} dx$
- Find the derivative of the function $f(x)=\sqrt{\sin^2x+e^x+1}$
- Help formulating sine function
- Trigonometry Word Problems
- Obtaining the absolute velocity of a moving train based on angle of raindrops with respect to vertical axis
- Step by step method to solve the following problem: find coordinates of B.
- A generator at a regional Power Station produces an alternating voltage, according to the function, 𝑉 below:
I don't think we can see the questions.
Thank you, I'm new to this site so I'm just figuring stuff out :)
This is a set of 7 problems, and may take about an hour to answer. I think you should offer a good bounty otherwise your post may not get answered.
I suggest you to ask your questions separately. You'll have a better chance of getting an answer.
Thank you so much you've been such a big help, if I wasn't in high school and severely broke I'd tip you😭 Wait actually lemme get some money quickly
Thank you for the coffee! I appreciate it. You may make money through Matchmaticians' affiliate program. See https://matchmaticians.com/affiliates