NEET Physics Solutions For Class 11 Chapter 10 Mathematical Tools – Trigonometry

NEET Physics Class 11 Chapter 10 Mathematical Tools – Trigonometry

Measurement Of Angle And Relationship Between Degrees And Radian

In navigation and astronomy, angles are measured in degrees, but in calculus, it is best to use units called radians because they simplify later calculations.

Let ACB be a central angle in a circle of radius r, as in the figure. Then the angle ACB or θ is defined in radius as –

NEET Physics Class 11 Notes Chapter 10 Mathematical Tools ACB Be A Central Angle In A Circle Of Radius R

θ = \(\frac{\text { Arc length }}{\text { Radius }}\)

θ = \(\frac{\widehat{A B}}{r}\)

If r = 1 then θ = AB

The radian measure for a circle of unit radius of angle ACB is defined as the length of the circular arc AB. Since the circumference of the circle is 2π and one complete revolution of a circle is 360º, the relation between radians and degrees is given by: π radians = 180º

Angle Conversion Formulas

1 degree = \(\frac{\pi}{180}\) (≈ 0.02) radian

Degrees to radians: multiply by \(\frac{\pi}{180}\)

1 radian ≈ 57 degrees

Radians to degrees : multiply by \(\frac{180}{\pi}\)

Question 1.

  1. Convert 45º to radians.
  2. Convert \(\frac{\pi}{6}\) rad to degrees.

Answer:

  1. \(\text { 45 – } \frac{\pi}{180}=\frac{\pi}{4} \mathrm{rad}\)
  2. \(\frac{\pi}{6} \cdot \frac{180}{\pi}=30^{\circ}\)

Question 2. Convert 30º to radians.

Answer:

⇒ \(30^{\circ} \times \frac{\pi}{180}\)

⇒ \(\frac{\pi}{6} \mathrm{rad}\)

Question 3. Convert \(\frac{\pi}{3}\) rad to degrees.

Answer:

⇒ \(\frac{\pi}{3} \times \frac{180}{\pi}\)= 60º

Standard values

  1. \(30^{\circ}=\frac{\pi}{6} \mathrm{rad}\)
  2. \(45^{\circ}=\frac{\pi}{4} \mathrm{rad}\)
  3. \(60^{\circ}=\frac{\pi}{3} \mathrm{rad}\)
  4. \(90^{\circ}=\frac{\pi}{2} \mathrm{rad}\)
  5. \(120^{\circ}=\frac{2 \pi}{3} \mathrm{rad}\)
  6. \(135^{\circ}=\frac{3 \pi}{4} \mathrm{rad}\)
  7. \(150^{\circ}=\frac{5 \pi}{6} \mathrm{rad}\)
  8. \(180^{\circ}=\pi \mathrm{rad}\)
  9. \(360^{\circ}=2 \pi \mathrm{rad}\)

(Check these values yourself to see that they satisfy the conversion formulae)

NEET Physics Class 11 Chapter 10 Mathematical Tools – Measurement Of Positive And Negative Angles

NEET Physics Class 11 Notes Chapter 10 Mathematical Tools Measurements Of Positive And Negetive Angles

An angle in the xy-plane is said to be in standard position if its vertex lies at the origin and its initial ray lies along the positive x-axis.

Angles measured counterclockwise from the positive x-axis are assigned positive measures; angles measured clockwise are assigned negative measures.

NEET Physics Class 11 Notes Chapter 10 Mathematical Tools Angles Measured Clockwise Are Assigned Negative Measures

NEET Physics Class 11 Chapter 10 Mathematical Tools – Six Basic Trigonometric Functions

The trigonometric function of a general angle θ is defined in terms of x, y, and r.

Sine: sinθ = \(\frac{\text { opp }}{\text { hyp }}=\frac{\mathrm{y}}{\mathrm{r}}\)

Cosecant: cosecθ = \(\frac{\text { hyp }}{\text { opp }}=\frac{\mathrm{r}}{\mathrm{y}}\)

NEET Physics Class 11 Notes Chapter 10 Mathematical Tools Six Trigonometric Functions

Cosine: cosθ = \(\frac{\text { adj }}{\text { hyp }}=\frac{x}{r}\)

Secant: secθ = \(\frac{\text { hyp }}{\text { adj }}=\frac{r}{x}\)

Tangent: tanθ = \(\frac{\text { opp }}{\text { adj }}=\frac{y}{x}\)

Cotangent: cotθ = \(\frac{\text { adj }}{\text { opp }}=\frac{x}{y}\)

Values Of Trigonometric Functions

If the circle has radius r = 1, the equations defining sinθ and cos θ become Cos θ = x, sinθ = y

We can then calculate the values of the cosine and sine directly from the coordinates of P.

Question 1. Find the six trigonometric ratios from the given figure

NEET Physics Class 11 Notes Chapter 10 Mathematical Tools The Six Trigonometric Ratios

Answer:

sinθ = \(\frac{\text { opp }}{\text { hyp }}=\frac{4}{5}\)

cosθ = \(\frac{\text { adj }}{\text { hyp }}=\frac{3}{5}\)

tan θ = \(\frac{\text { opp }}{\text { adj }}=\frac{4}{3}\)

cosec θ = \(\frac{\text { hyp }}{\text { opp }}=\frac{5}{4}\)

sec θ = \(\frac{\text { hyp }}{\text { adj }}=\frac{5}{3}\)

cotθ = \(\frac{\text { adj }}{\text { opp }}=\frac{3}{4}\)

Question 2. Find the sine and cosine of angle θ shown in the unit circle if the coordinates of point p are as shown.

Answer:

NEET Physics Class 11 Notes Chapter 10 Mathematical Tools The Sine And Cosine Of Angle Theta

cos θ = x-coordinate of P = – \(\frac{1}{2}\)

2sin θ = y-coordinate of P = \(\frac{\sqrt{3}}{2}\)

NEET Physics Class 11 Chapter 10 Mathematical Tools – Rules For Finding Trigonometric Ratio Of Angles Greater Than 90°

Step 1 → Identify the quadrant in which the angle lies.

Step 2 → If angle = (nπ ± θ) where n is an integer. Then trigonometric function of (nπ ± θ)= same trigonometric function of θ and the sign will be decided by the CAST Rule.

NEET Physics Class 11 Notes Chapter 10 Mathematical Tools The Cast Rule

If angle = \(\left[(2 n+1) \frac{\pi}{2} \pm \theta\right]\) where n is an integer. Then the trigonometric function of \(\left[(2 n+1) \frac{\pi}{2} \pm \theta\right]\)= complimentary trigonometric function of θ and sign will be decided by CAST Rule.

Values of sin θ, cos θ, and tan θ for some standard angles.

NEET Physics Class 11 Notes Chapter 10 Mathematical Tools Values Of Some Standard Angles

Question 1. Evaluate sin 120°

Answer:

sin 120°

= sin (90° + 30°)

= cos 30°

⇒ \(\frac{\sqrt{3}}{2}\)

Aliter sin 120°

= sin (180° – 60°)

= sin 60°

⇒ \(\frac{\sqrt{3}}{2}\)

Question 2. Evaluate cos 135°

Answer:

cos 135°

= cos (90° + 45°)

= – sin 45°

⇒ \(-\frac{1}{\sqrt{2}}\)

Question 3. Evaluate cos 210°

Answer:

cos 210°

= cos (180° + 30°)

= – cos30°

⇒ \(-\frac{\sqrt{3}}{2}\)

Question 4. Evaluate tan 210°

Answer:

tan 210°

= tan (180° + 30°)

= tan 30°

= \(\frac{1}{\sqrt{3}}\)

NEET Physics Class 11 Chapter 10 Mathematical Tools – General Trigonometric Formulas

Question 1. \(\cos ^2 \theta+\sin ^2 \theta=1\)

Answer:

⇒ \(1+\tan ^2 \theta=\sec ^2 \theta\)

⇒ \(1+\cot ^2 \theta={cosec}^2 \theta\)

Question 2. cos(A + B) = cos A cos B – sin A sin B

Answer:

sin( A + B) = sin A cos B + cos A sin B

⇒ \(\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}\)

Question 3. sin 2θ = 2 sin θ cos θ; cos 2θ = cos2 θ – sin2θ = 2cos2 θ – 1 = 1 – 2sin2 θ

Answer:

⇒ \(\cos ^2 \theta=\frac{1+\cos 2 \theta}{2}\)

⇒ \(\sin ^2 \theta=\frac{1-\cos 2 \theta}{2}\)

Question 4.

NEET Physics Class 11 Notes Chapter 10 Mathematical Tools In Triangle ABC Sine Rule

In Δ ABC, the sine rule

Answer:

ΔABC need not be right-angled, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)

Question 5. Cosine Rule:

Answer:

cosA = \(\frac{\mathrm{b}^2+\mathrm{C}^2-\mathrm{a}^2}{2 \mathrm{bc}}\)

cosB = \(\frac{a^2+C^2-b^2}{2 a c}\)

cosC = \(\frac{a^2+b^2-c^2}{2 a b}\)

NEET Physics Class 11 Chapter 10 Mathematical Tools – Coordinate Geometry

To specify the position of a point in space, we use a right-handed rectangular axes coordinate system. This system consists of

  1. Origin
  2. Axis or axes.

If a point is known to be on a given line or in a particular direction only one coordinate is necessary to specify its position, if it is in a plane, two coordinates are required, if it is in space three coordinates are needed.

Origin

This is any fixed point that is convenient to you. All measurements are taken w.r.t. this fixed point.

Axis or Axes

Any fixed direction passing through the origin and convenient to you can be taken as an axis.

  • If the position of a point or the position of all the points under consideration always happens to be in a particular direction, then only one axis is required. This is generally called the x-axis.
  • If the positions of all the points under consideration are always in a plane, two perpendicular axes are required.
  • These are generally called the x and y-axis. If the points are distributed in space, three perpendicular axes are taken which are called the x, y, and z-axis.

Position Of A Point In xy Plane

The position of a point is specified by its distances from the origin along (or parallel to) the x and y-axis as shown in the figure.

NEET Physics Class 11 Notes Chapter 10 Mathematical Tools Position Of A Point In Xy Plane

Here x-coordinate and y-coordinate are called abscissa and ordinate respectively.

Distance Formula

The distance between two points (x1, y1) and (x2, y2) is given by

d = \(\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}\)

Note: In space d = \(\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}\)

Slope Of A Line

The slope of a line joining two points A(x1, y1) and B(x2, y2) is denoted by
m and is given by

m = \(\frac{\Delta \mathrm{y}}{\Delta \mathrm{x}}=\frac{\mathrm{y}_2-\mathrm{y}_1}{\mathrm{x}_2-\mathrm{x}_1}=\tan \theta\) [If both axes have identical scales]

NEET Physics Class 11 Notes Chapter 10 Mathematical Tools Slope Of A Line

Here θ is the angle made by a line with a positive x-axis. The slope of a line is a quantitative measure of inclination.

Question 1. For points (2, 14) find abscissa and ordinate. Also, find the distance from the y and x-axis.

Answer:

Abscissa = x-coordinate = 2 = distance from y-axis.

Ordinate = y-coordinate = 14 = distance from the x-axis.

Question 2. Find the value of a if distances between the points (–9 cm, a cm) and (3 cm, 3cm) is 13 cm.

Answer:

By using distance formula d = \(\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}\)

⇒ \(13 \sqrt{[3-(-9)]^2+[3-a]^2}\)

⇒ 132 = 122 + (3 – a)2

⇒ (3 – a)2 = 132 – 122 = 52

⇒ (3 – a) = ± 5

⇒ a = 2 cm or 8 cm

Question 3. A dog wants to catch a cat. The dog follows the path whose equation is y-x = 0 while the cat follows the path whose equation is x2 + y2 = 8. The coordinates of possible points for catching the cat are.

  1. (2, – 2)
  2. (2, 2)
  3. (–2, 2)
  4. (–2, 2)
  5. (2, 4)

Answer:

Let catching point be (x1, y1) then, y1 – x1 = 0 and x12 + y12 = 8

Therefore, 2x12 = 8

⇒ x12 = 4

⇒ x1 = ± 2;

so possible ae (2, 2) and (–2, –2).

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