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This is a PDF of class 11th trigonometry formulas for class 11th students in CBSE and other boards.
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Class 11 maths trigonometry questions for for exam
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Some examples from this set of practice questions
1.
### Question: Using the double angle formula for sine, calculate \\(\\sin 2A\\) if \\(\\sin A = \\frac{3}{5}\\) and \\(A\\) is in the first quadrant.
Answer: Given \\(\\sin A = \\frac{3}{5}\\) and knowing \\(A\\) is in the first quadrant, we can find \\(\\cos A\\) using the Pythagorean identity. 1. First, find \\(\\cos A\\): \\[ \\sin^2 A + \\cos^2 A = 1 \\] \\[ \\left(\\frac{3}{5}\\right)^2 + \\cos^2 A = 1 \\] \\[ \\frac{9}{25} + \\cos^2 A = 1 \\] \\[ \\cos^2 A = 1 - \\frac{9}{25} \\] \\[ \\cos^2 A = \\frac{25}{25} - \\frac{9}{25} \\] \\[ \\cos^2 A = \\frac{16}{25} \\] \\[ \\cos A = \\sqrt{\\frac{16}{25}} = \\frac{4}{5} \\] Since \\(A\\) is in the first quadrant, \\(\\cos A\\) is positive, so \\(\\cos A = \\frac{4}{5}\\). 2. Now, use the double angle formula for sine: \\[ \\sin 2A = 2 \\sin A \\cos A \\] Substitute the values: \\[ \\sin 2A = 2 \\left(\\frac{3}{5}\\right) \\left(\\frac{4}{5}\\right) \\] \\[ \\sin 2A = 2 \\left(\\frac{12}{25}\\right) \\] \\[ \\sin 2A = \\frac{24}{25} \\] Thus, \\(\\sin 2A = \\frac{24}{25}\\).
2.
Prove that \\(\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B\\).
Answer: Using the angle addition formula for sine, we have: \\[\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B\\] This is a standard trigonometric identity.
3.
Simplify \\(\\sec^2 x - \\tan^2 x\\).
Answer: Using the Pythagorean identity: \\[\\sec^2 x - \\tan^2 x = 1\\]
4.
If \\(\\sin \\theta = \\frac{3}{5}\\), find \\(\\cos \\theta\\).
Answer: Using \\(\\sin^2 \\theta + \\cos^2 \\theta = 1\\), \\[\\left(\\frac{3}{5}\\right)^2 + \\cos^2 \\theta = 1\\] \\[\\frac{9}{25} + \\cos^2 \\theta = 1\\] \\[\\cos^2 \\theta = 1 - \\frac{9}{25} = \\frac{16}{25}\\] \\[\\cos \\theta = \\pm \\frac{4}{5}\\]
5.
Prove that \\(\\cos(A - B) = \\cos A \\cos B + \\sin A \\sin B\\).
Answer: Using the angle subtraction formula for cosine, we have: \\[\\cos(A - B) = \\cos A \\cos B + \\sin A \\sin B\\] This is another standard trigonometric identity.
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Class 11th trigonometry important formulasHow to use this template
, ### Basic Trigonometric Ratios
For an angle \(\theta\) in a right-angled triangle:
- \(\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
- \(\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
- \(\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin\theta}{\cos\theta}\)
- \(\csc\theta = \frac{1}{\sin\theta} = \frac{\text{Hypotenuse}}{\text{Opposite}}\)
- \(\sec\theta = \frac{1}{\cos\theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}}\)
- \(\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta} = \frac{\text{Adjacent}}{\text{Opposite}}\)
### Trigonometric Identities
1. **Pythagorean Identities:**
- \(\sin^2\theta + \cos^2\theta = 1\)
- \(1 + \tan^2\theta = \sec^2\theta\)
- \(1 + \cot^2\theta = \csc^2\theta\)
2. **Sum and Difference Formulas:**
- \(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\)
- \(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\)
- \(\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}\)
3. **Double Angle Formulas:**
- \(\sin 2A = 2 \sin A \cos A\)
- \(\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A\)
- \(\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}\)
4. **Half Angle Formulas:**
- \(\sin \frac{A}{2} = \pm \sqrt{\frac{1 - \cos A}{2}}\)
- \(\cos \frac{A}{2} = \pm \sqrt{\frac{1 + \cos A}{2}}\)
- \(\tan \frac{A}{2} = \pm \sqrt{\frac{1 - \cos A}{1 + \cos A}} = \frac{\sin A}{1 + \cos A} = \frac{1 - \cos A}{\sin A}\)
### Product to Sum Formulas
- \(\sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)]\)
- \(\cos A \cos B = \frac{1}{2} [\cos(A + B) + \cos(A - B)]\)
- \(\sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)]\)
### Sum to Product Formulas
- \(\sin A + \sin B = 2 \sin \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)\)
- \(\sin A - \sin B = 2 \cos \left(\frac{A + B}{2}\right) \sin \left(\frac{A - B}{2}\right)\)
- \(\cos A + \cos B = 2 \cos \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)\)
- \(\cos A - \cos B = -2 \sin \left(\frac{A + B}{2}\right) \sin \left(\frac{A - B}{2}\right)\)
### Inverse Trigonometric Functions
- \(\sin^{-1} x\), \(\cos^{-1} x\), \(\tan^{-1} x\), \(\csc^{-1} x\), \(\sec^{-1} x\), \(\cot^{-1} x\)
### Important Values of Trigonometric Functions
For key angles \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\):
- \(\sin 0^\circ = 0\), \(\sin 30^\circ = \frac{1}{2}\), \(\sin 45^\circ = \frac{1}{\sqrt{2}}\), \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), \(\s
- \(\cos 0^\circ = 1\), \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), \(\cos 45^\circ = \frac{1}{\sqrt{2}}\), \(\cos 60^\circ = \frac{1}{2}\),
- \(\tan 0^\circ = 0\), \(\tan 30^\circ = \frac{1}{\sqrt{3}}\), \(\tan 45^\circ = 1\), \(\tan 60^\circ = \sqrt{3}\), \(\tan 90^\circ\) (un
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