Download App

Sine and Cosine Formulae and Their Applications — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSine and Cosine Formulae and Their Applications4 Marks
Question
If in a $\triangle\text{ABC},\cos^2\text{A}+\cos^2\text{B}+\cos^2\text{C}=1,$ prove that the triangle is right angled.
✓

Answer

Let ABC be any triangle. In $\triangle\text{ABC},$ $\cos^2\text{A}+\cos^2\text{B}+\cos^2\text{C}=1$ $\Rightarrow\cos^2\text{A}+\cos^2\text{B}+\cos^2[\pi-(\text{B + A})]=1$ $(\because\text{A + B + C} = \pi)$ $\Rightarrow\cos^2\text{A}+\cos^2\text{B}+\cos^2(\text{B + A})=1$ $\Rightarrow\cos^2\text{A}+\cos^2\text{B}=1-\cos^2(\text{B + A})$ $\Rightarrow\cos^2\text{A}+\cos^2\text{B}=\sin^2(\text{B + A})$ $\Rightarrow\cos^2\text{A}+\cos^2\text{B}=(\sin\text{A}\cos\text{B}+\cos\text{A}\sin\text{B})^2$ $\Rightarrow\cos^2\text{A}+\cos^2\text{B}=\sin^2\text{A}\cos^2\text{B}+\cos^2\text{A}\sin^2\text{B}\\+2\sin\text{A}\sin\text{B}\cos\text{A}\cos\text{B}$ $\Rightarrow\cos^2\text{A}(1-\sin^2\text{B})+\cos^2\text{B}(1-\sin^2\text{A})$ $=2\sin\text{A}\sin\text{B}\cos\text{A}\cos\text{B}$ $\Rightarrow2\cos^2\text{A}\cos^2\text{B}=2\sin\text{A}\sin\text{B}\cos\text{A}\cos\text{B}$ $\Rightarrow\cos\text{A}\cos\text{B}=\sin\text{A}\sin\text{B}$ $\Rightarrow\cos\text{A}\cos\text{B}-\sin\text{A}\sin\text{B}=0$ $\Rightarrow\cos(\text{A + B)}=0$ $\Rightarrow\cos\text{(A + B)}=\cos90^{\circ}$ $\Rightarrow\text{A + B}=90^{\circ}$ $\Rightarrow\text{C}=90^{\circ}$ $(\because\text{A + B + C }= 180^{\circ})$ Hence, $\triangle\text{ABC}$ is right angled.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 4 marks)" from Sine and Cosine Formulae and Their Applications→Sine and Cosine Formulae and Their Applications — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
The number of people who read at least one of the newspapers.
→
If a, b, c, are in G.P., prove that the following are also in G.P. $\text{a}^3,\text{b}^3,\text{c}^3$
→
If ${^\text{16}}\text{C}_{\text{r}}={^\text{16}}\text{C}_{\text{r+2}},$ find ${^\text{7}}\text{C}_{4}.$
→
Find the equation of the hyperbola whoseFoci are (6, 4) and (-4, 4) and eccentricity is 2.
→
How many permutations can be formed by the letters of the word, 'VOWELS', when.
  1. There is no restriction on letters?
  2. Each word begins with E?
  3. Each word begins with O and ends with L?
  4. All vowels come together?
  5. All consonants come together?
→
How many words can be formed by taking 4 letters at a time from the letters of the word 'MORADABAD'?
→
Find the mean variance and standard deviation for the following data: $6, 7, 10, 12, 13, 4, 8, 12.$
→
Prove the following by the principle of mathematical induction: $1 + 3 + 5 + ... + (2n - 1) = n^2$ i.e., the sum of first n odd natural numbers is $n^2​​​​​​​$.
→
Find the angles between the following pairs of straight lines: 3x - y + 5 = 0 and x - 3y + 1 = 0
→
The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and $100m$ long is supported by vertical wires attached to the cable, the longest wire being $30m$ and the shortest being 6m. Find the length of a supporting wire attached to the roadway 18m from the middle.
→