Question

If sin 17° = 7, then the value of
(sec 17° - sin 73°) is​

Answer 1

Answer:

Given that

$sin17 \degree = 7 \\ so \\ {sin}^{2} 17 \degree = 49$

As we know that,

${sin}^{2} \theta + {cos}^{2} \theta = 1$

So,

${cos}^{2} 17 \degree = 1 - 49 \\ = - 48 \: \\ \implies \: cos17\degree = \sqrt{ - 48}$

Now,

$as \: \: \: cos17\degree = \sqrt{ - 48} \\ cos(90 - 73) = \sqrt{ - 48} \\ sec \: 17 = \frac{1}{ \sqrt{ - 48} }$

As, cos (90 - θ) = sinθ

So,

$cos(90 - 73) = sin73 \degree$

So,

$sin73 \degree = \sqrt{ - 48}$

Now,

$sec17 - sin73 \\ = \frac{1}{ \sqrt{ - 48} } - \sqrt{ - 48}$

$= \dfrac{1}{ \sqrt{48} \iota} - \sqrt{48} \iota \\ \\ = \dfrac{1 - 48 { \iota}^{2} }{ \sqrt{48} \iota} \\ \\ = \frac{1 + 48}{ \sqrt{48} \iota } \\ \\ = \frac{49}{ \sqrt{48}\iota} \times \frac{ - \sqrt{48}\iota}{ - \sqrt{48}\iota} \\ \\ = \frac{ - 49 \sqrt{48} \iota}{ - 48 {\iota}^{2} } \\ \\ = \frac{ - 49 \sqrt{48} \iota}{48}$

OR

As we know that,

sin(90-θ) = cosθ

so,

sin 73 = cos 17.

So ,

sec17 - cos 17

= sec17 - (1/sec17)

=(sec^2 17 - 1) /(sec17)

$= \dfrac{ {tan}^{2} 17}{sec17} \\ \\ = \dfrac{ \dfrac{ {sin}^{2}17 }{ {cos}^{2}17 } }{ \dfrac{1}{cos17} } \\ \\ = \frac{{sin}^{2} 17}{cos17} \\ \\ = sin17 \times \frac{sin17}{cos17} \\ \\ = 7 \times tan17 \\ \\ = 7tan17$

Answer 2

Answer:

Basic Formulas:

• Sin A = Cos ( 90 - A )
• Sec²A - 1 = Tan²A

Given:

• Sin 17° = 7

To find:

• ( Sec 17° - Sin 73° )

Solution:

Using the first formula we can write Sin 73° as follows:

⇒ Sin 73° = Cos ( 90 - 73 )°

⇒ Sin 73° = Cos 17°

Substituting it there, we get,

⇒ Sec 17° - Cos 17°

Now we can write Cos 17° as 1 / Sec 17°. By this we get,

⇒ Sec 17° - ( 1 / Sec 17° )

Taking LCM we get,

⇒ ( Sec²17° - 1 ) / Sec 17°

Now applying Second formula we get,

⇒ Tan² 17° / Sec 17°

Writing Tan and Sec values in terms of Sin and Cos we get,

⇒ ( Sin² 17° / Cos²17° ) ÷ ( 1 / Cos 17° )

⇒ ( Sin² 17° / Cos 17° )

⇒ Sin 17° × Sin 17° / Cos 17°

⇒ Sin 17° × Tan 17°

Simplifying further we get,

⇒ 7 × Tan 17°

This is the value. Unless we know the values, we cant arrive at a defined value. Another method is by using Complex Numbers as posted above.