Trigonometric Functions Exercise 02
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:
 sin x/2 = sqrt(3)/2
 cos x/2 = 1/2
 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:

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

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

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:

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

Next, find the value of sinx using the equation: sinx = √(1cos^2x) sinx = √(1(5/13)^2) sinx = √(125/169) sinx = √(144/169) sinx = 12/13

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

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

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:

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

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

Secx = 1/Cosx Secx = 1/(1/2) Secx = 2

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

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:

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

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:

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

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

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

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! + …