Advertisements
Advertisements
Question
Eliminate θ from the equations a sec θ – c tan θ = b and b sec θ + d tan θ = c
Solution
Taking sec θ = x and tan θ = y we get the equations as
ax – cy = b .....(1)
bx + dy = c .....(2)
(1) × d ⇒ adx – cdy = bd ....(3)
(2) × c ⇒ bcx + cdy = c2 ......(4)
(3) + (4) ⇒ x(ad + bc) = bd + c2
∴ x = `("bd" + "c"^2)/("ad" + "bc")`
(i.e) sec θ = `("bd" "c")/("ad" "bc")`
(2) × a ⇒ ax + ay = ac ......(5)
Now (1) × b ⇒ abx – bcy = b2 ......(6)
(5) – (6) ⇒ y(ad + bc) = ac – b2
⇒ y = `("ac" - "b"^2)/("ad" + "bc")`
(i.e) tan θ = `("ac" - "b"^2)/("ad" + "bc")`
∴ sec θ = `("bd" + "c"^2)/("ad" + "bc")` and tan θ = `("ac" + "b"^2)/("ad" + "bc")`
We know sec2θ – tan2θ = 1
⇒ `(("bd" + "c"^2)^2)/(("ad" + "bc")^2) - (("ac" - "b"^2)^2)/(("ad" + "bc")^2` = 1
⇒ `(("bd" + "c"^2)^2 - ("ac" - "b"^2)^2)/(("ad" + "bc")^2` = 1
⇒ (bd+ c2)2 – (ac – b2)2 = (ad +bc)2
⇒ (bd+ c2)2 – (ad + bc)2 = (ac +b2)2
APPEARS IN
RELATED QUESTIONS
Identify the quadrant in which an angle given measure lies
25°
Identify the quadrant in which an angle given measure lies
825°
Identify the quadrant in which an angle given measure lies
– 55°
For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
395°
For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
525°
For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
1150°
For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
– 270°
For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
– 450°
If a cos θ − b sin θ = c, show that a sin θ + b cos θ = `+- sqrt("a"^2 + "b"^2 - "c"^2)`
If sin θ + cos θ = m, show that cos6θ + sin6θ = `(4 - 3("m"^2 - 1)^2)/4`, where m2 ≤ 2
If `(cos^4alpha)/(cos^2beta) + (sin^4alpha)/(sin^2beta)` = 1, prove that `(cos^4beta)/(cos^2alpha) + (sin^4beta)/(sin^2alpha)` = 1
If x = `sum_("n" = 0)^oo cos^(2"n") theta, y = sum_("n" = 0)^oo sin^(2"n") theta` and z = `sum_("n" = 0)^oo cos^(2"n") theta, sin^(2"n") theta, 0 < theta < pi/2`, then show that xyz = x + y + z. [Hint: Use the formula 1 + x + x2 + x3 + . . . = `1/(1 - x), where |x| < 1]
If tan2 θ = 1 – k2, show that sec θ + tan3 θ cosec θ = (2 – k2)3/2. Also, find the values of k for which this result holds
If cosec θ – sin θ = a3 and sec θ – cos θ = b3, then prove that a2b2 (a2 + b2) = 1
Choose the correct alternative:
The maximum value of `4sin^2x + 3cos^2x + sin x/2 + cos x/2` is
Choose the correct alternative:
Which of the following is not true?
