Question:

Find the principal and general solutions of the following equations:(i) tanx=√3 ​(ii) secx=2 (iii) cotx=−√3 ​(iv) cosecx=−2

Answer:

(i) Principal Solution: x = π/3 + 2πk, where k is any integer General Solution: x = π/3 + 2πk ± 2πn, where k and n are any integers

(ii) Principal Solution: x = π/6 + πk, where k is any integer General Solution: x = π/6 + πk ± πn, where k and n are any integers

(iii) Principal Solution: x = −π/3 + 2πk, where k is any integer General Solution: x = −π/3 + 2πk ± 2πn, where k and n are any integers

(iv) Principal Solution: x = 5π/6 + πk, where k is any integer General Solution: x = 5π/6 + πk ± πn, where k and n are any integers

Question:

tanx=−4​/3, x in quadrant II. Find the value of sin x​/2,cos x​/2,tan x/2

Answer:

1. sin x/2 = -sqrt(3)/2
2. cos x/2 = 1/2
3. tan x/2 = -sqrt(3)/3

Question:

Find the values of other five trigonometric functions if sin x = 3​/5, x lies in second quadrant.

Answer:

Step 1: Since x lies in the second quadrant, the value of x will be in the range of 90° to 180°.

Step 2: Use the trigonometric identity to find the value of cos x, cos x = 4/5

Step 3: Use the trigonometric identity to find the value of tan x, tan x = 3/4

Step 4: Use the trigonometric identity to find the value of cosec x, cosec x = 5/3

Step 5: Use the trigonometric identity to find the value of sec x, sec x = 5/4

Step 6: Use the trigonometric identity to find the value of cot x, cot x = 4/3

Question:

Value of cosec (−1410°) is A 1​/2 B −1​/2 C √3/2 ​​D 2

Answer:

Answer: B -1/2

Explanation:

To solve this problem, we need to use the definition of cosecant. Cosecant is the reciprocal of the sine of an angle.

Therefore, we can calculate the value of cosecant (−1410°) as follows:

cosec (−1410°) = 1 / sin (−1410°)

Since sin (−1410°) = -1/2,

cosec (−1410°) = -1/2

Hence, the answer is B -1/2.

Question:

Find the value of other five trigonometric ratios: When sinx=3​/5, and x lies in second quadrant.

Answer:

Step 1: Find the value of cosx.

cosx = 4/5

Step 2: Find the value of tanx.

tanx = -3/4

Step 3: Find the value of cosecx.

cosecx = -5/3

Step 4: Find the value of secx.

secx = -4/3

Step 5: Find the value of cotx.

cotx = -4/3

Question:

cosx=−1​/3, x in quadrant III. Find the value of sin x​/2,cos x​/2,tan x​/2

Answer:

1. sin x/2 = √(1 - cos2x/4) = √(1 - (-1/9)) = √(10/9)

2. cos x/2 = √(1 - sin2x/4) = √(1 - (-1/9)) = √(10/9)

3. tan x/2 = sin x/2/cos x/2 = √(10/9)/√(10/9) = 1

Question:

Find the value of other five trigonometric ratios: secx=13​/5, x lies in fourth quadrant.

Answer:

1. First, find the value of cosx using the equation: cosx = 1/secx cosx = 5/13

2. Next, find the value of sinx using the equation: sinx = √(1-cos^2x) sinx = √(1-(5/13)^2) sinx = √(1-25/169) sinx = √(144/169) sinx = 12/13

3. Then, find the value of cosecx using the equation: cosecx = 1/sinx cosecx = 13/12

4. Next, find the value of cotx using the equation: cotx = cosx/sinx cotx = 5/12

5. Finally, find the value of tanx using the equation: tanx = sinx/cosx tanx = 12/5

Question:

Find the value of tan19π/3

Answer:

Step 1: Convert 19π/3 into radians.

19π/3 = 19π/3 × (180°/π) = 570°

Step 2: Find the value of tan 570°.

tan 570° = tan(540° + 30°) = tan 540° × tan 30° + tan 540° ÷ tan 30° = 0 × 1/√3 + 0 = 0

Question:

sinx=1​/4, x in quadrant II. Find the value of sinx/2

Answer:

Step 1: Find the value of x in quadrant II that satisfies the equation ‘sinx=1/4’.

Using the inverse sine function, x = arcsin(1/4) = 0.92729522

Step 2: Calculate sinx/2.

sinx/2 = sin(0.92729522)/2 = 0.46364761

Question:

Find the value of other five trigonometric ratios: cosx=−1​/2, x lies in third quadrant.

Answer:

1. Sinx = √(1 - (cosx)^2) Sinx = √(1 - (-1/2)^2) Sinx = √(1 - 1/4) Sinx = √3/2

2. Tanx = Sinx/Cosx Tanx = √3/2 / (-1/2) Tanx = -√3

3. Secx = 1/Cosx Secx = 1/(-1/2) Secx = -2

4. Cosecx = 1/Sinx Cosecx = 1/√3/2 Cosecx = 2/√3

5. Cotx = Cosx/Sinx Cotx = (-1/2)/√3/2 Cotx = -2/√3

Question:

Find the value of other five trigonometric ratios: cotx=3​/4, x lies in third quadrant.

Answer:

1. First, we need to find the measure of angle x. To do this, we can use the inverse cotangent function: x = arccot(3/4)

2. Next, we can find the value of the other five trigonometric ratios for angle x: sin x = -sqrt(3)/2 cos x = -1/2 tan x = -sqrt(3) sec x = -2 csc x = -2sqrt(3)

Question:

tanx=−5​/12,x lies in second quadrant.

Answer:

1. First, use the Pythagorean Theorem to determine the length of the adjacent side: adjacent side = 12

2. Next, use the tangent ratio to calculate the opposite side: opposite side = -5

3. Then, use the inverse tangent function to calculate the angle: angle = -63.43°

4. Since the angle lies in the second quadrant, the angle must be between -90° and 0°: x = -63.43°

Question:

Find the value of sin(765o).

Answer:

Step 1: Convert 765° into radians.

1° = π/180 radians

765° = (765 × π) / 180 radians

Step 2: Use the formula sin(x) = x - x3/3! + x5/5! - x7/7! + … to calculate the value of sin(765o).

sin(765o) = (765π/180) - (765π/180)3/3! + (765π/180)5/5! - (765π/180)7/7! + …