Question

1. If tan A = 1, show that sin A cos A = 1/2 (Using Pyth )​

Answer 1

Solution :-

Given ,

• tanA = 1

We need to show that ,

• sinA cosA = 1/2 [ Using Pythagoras theorem ]

If tanA = 1 height of triangle is 1 & base of triangle is 1 .

Now , finding hypotenuse using Pythagoras theorem .

Hyp² = Base² + Height²

→ Hyp² = 1² + 1²

→ Hyp² = 1 + 1

→ Hyp² = 2

→ Hypotenuse = √2

• sinA = opposite/Hypotenuse

• cosA = Adjacent/Hypotenuse

♦ sinA = 1/√2

♦ cosA = 1/√2

Now finding, sinA cosA

⇒ 1/√2 × 1/√2

⇒ 1/(√2)²

⇒ sinA cosA = 1/2

Hence , sinA cosA = 1/2

Answer 2

♣ Qᴜᴇꜱᴛɪᴏɴ :

• If tan A = 1, show that sin A cos A = $\sf{\dfrac{1}{2}}$  (Using Pythagoras Theorem )​

★═════════════════★

♣ ᴛᴏ ᴘʀᴏᴠᴇ :

sin A cos A = $\sf{\dfrac{1}{2}}$  (Using Pythagoras Theorem )​

★═════════════════★

♣ ᴀɴꜱᴡᴇʀ :

$\setlength{\unitlength}{.4in} \begin{picture}(7,5)(0,0) \linethickness{1pt} \put(0,0){\line(1,0){4}} \put(4,0){\line(0,1){3}} \put(0,0){\line(4,3){4}} \put(2,-.25){\makebox(0,-1){\sf{Base}}} \put(4.25,1.5){\makebox(2,0){\sf{Height}}} \put(2,2){\makebox(-2,0){\bf{Hypotenuse}}} \end{picture}$

We know :

$\large\boxed{\sf{tan A = \dfrac{Height}{Base}}}$

Also :

$\boxed{\sf{sinA=\dfrac{Height}{Hypotenuse}}}$

$\boxed{\sf{cosA=\dfrac{Base}{Hypotenuse}}}$

$\setlength{\unitlength}{.4in} \begin{picture}(7,5)(0,0) \linethickness{1pt} \put(0,0){\line(1,0){4}} \put(4,0){\line(0,1){3}} \put(0,0){\line(4,3){4}} \put(2,-.25){\makebox(0,-1){\sf{1}}} \put(4.25,1.5){\makebox(0.5,0){\sf{1}}} \put(2,2){\makebox(-2,0){\bf{Hypotenuse}}} \end{picture}$

If tanA = 1

• Base = 1
• Height = 1

According to Pythagoras Theorem :

Hypotenuse² = Height² + Base²

According To Question :

Hypotenuse² = 1² + 1²

Hypotenuse² = 1 + 1

Hypotenuse² = 2

Hypotenuse = √2

Now we need to Prove sin A cos A = $\sf{\dfrac{1}{2}}$

→ Taking L.H.S

sin A cos A

$=\sf{\dfrac{Height}{Hypotenuse}\:\times \:\dfrac{Base}{Hypotenuse}}$

$=\sf{\dfrac{1}{\sqrt{2}}\:\times\:\dfrac{1}{\sqrt{2}}}$

$=\sf{\dfrac{1}{\sqrt{2}^2}}$

$=\sf{\dfrac{1}{2}}$

= R.H.S

★  Hence Proved ★

★═════════════════★

♣ ᴏᴛʜᴇʀ ᴛʀɪɢᴏɴᴏᴍᴇᴛʀɪᴄ ɪᴅᴇɴᴛɪᴛɪᴇꜱ  :

• sin(θ) = Height / Hypotenuse
• cos(θ) = Base / Hypotenuse
• tan(θ) = Height / Base
• cosec(θ) =  Hypotenuse  / Height
• sec(θ) =  Hypotenuse  / Base  
• tan(θ) =  Base / Height