Question

write the trinometric identities

Answer 1

$\mathfrak{dear\:user}$

$\mathfrak{question\:write \:the \:trinometric \:identities}$

$\mathfrak{here\:is\;the\;solution}$

$\mathbb{ANSWER}$

$\boxed{\begin{minipage}{6cm} Important Trigonometric identities :- \\ \\ \: \: 1)\:\sin^2\theta+\cos^2\theta=1 \\ \\ 2)\:\sin^2\theta= 1-\cos^2\theta \\ \\ 3)\:\cos^2\theta=1-\sin^2\theta \\ \\ 4)\:1+\cot^2\theta=\text{cosec}^2 \, \theta \\ \\5)\: \text{cosec}^2 \, \theta-\cot^2\theta =1 \\ \\ 6)\:\text{cosec}^2 \, \theta= 1+\cot^2\theta \\\ \\ 7)\:\sec^2\theta=1+\tan^2\theta \\ \\ 8)\:\sec^2\theta-\tan^2\theta=1 \\ \\ 9)\:\tan^2\theta=\sec^2\theta-1 \end{minipage}}$

$\mathbb{EXTRA INFORMATION}$

$\bullet\:\sf Trigonometric\:Values :\\\\\boxed{\begin{tabular}{c|c|c|c|c|c}Radians/Angle & 0 & 30 & 45 & 60 & 90\\\cline{1-6}Sin \theta & 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} & 1\\\cline{1-6}Cos \theta & 1 & \dfrac{\sqrt{3}}{2}& \dfrac{1}{\sqrt{2}}& \dfrac{1}{2}&0\\\cline{1-6}Tan \theta&0& \dfrac{1}{\sqrt{3}}&1&\sqrt{3}&Not D{e}fined\end{tabular}}$

$\to PLEASE\;DON'T\:COPY$

$\mathcal{HOPE\:IT\;HELPS}$

$\mathcal{BY \:BRAINLY\;ROSHAN}$

Answer 2

ꜰɪʀꜱᴛ ᴏꜰ ᴀʟʟ, ɪɴ ᴛʀɪɢᴏɴᴏᴍᴇᴛʀʏ, ᴡᴇ ʜᴀᴠᴇ ᴛʀɪɢᴏɴᴏᴍᴇᴛʀɪᴄ ᴇQᴜᴀᴛɪᴏɴꜱ. ʟɪᴋᴇ ᴀʟɢᴇʙʀᴀɪᴄ ᴇQᴜᴀᴛɪᴏɴꜱ, ꜱᴏʟᴠɪɴɢ ᴀ ᴛʀɪɢᴏɴᴏᴍᴇᴛʀɪᴄ ᴇQᴜᴀᴛɪᴏɴ ɪꜱ ᴛʜᴇ ᴘʀᴏᴄᴇꜱꜱ ᴏꜰ ꜰɪɴᴅɪɴɢ ᴛʜᴏꜱᴇ ᴠᴀʟᴜᴇꜱ ᴏꜰ ᴛʜᴇ ᴠᴀʀɪᴀʙʟᴇ ᴛʜᴀᴛ ᴍᴀᴋᴇ ᴛʜᴇ ᴇQᴜᴀᴛɪᴏɴ ᴀ ᴛʀᴜᴇ ꜱᴛᴀᴛᴇᴍᴇɴᴛ, ᴀɴᴅ, ʟɪᴋᴇ ᴍᴀɴʏ ᴀʟɢᴇʙʀᴀɪᴄ ᴇQᴜᴀᴛɪᴏɴꜱ, ᴍᴀɴʏ ᴛʀɪɢᴏɴᴏᴍᴇᴛʀɪᴄ ᴇQᴜᴀᴛɪᴏɴꜱ ᴀʀᴇ ᴄᴏɴᴅɪᴛɪᴏɴᴀʟ ᴡʜɪᴄʜ ᴍᴇᴀɴꜱ ᴛʜᴀᴛ ᴏᴜᴛ ᴏꜰ ᴛʜᴇ ꜱᴇᴛ ᴏꜰ ᴀʟʟ ᴛʜᴇ ᴘᴏꜱꜱɪʙʟᴇ ʀᴇᴘʟᴀᴄᴇᴍᴇɴᴛꜱ ꜰᴏʀ ᴛʜᴇ ᴠᴀʀɪᴀʙʟᴇ ᴏɴʟʏ ᴀ ᴄᴇʀᴛᴀɪɴ ᴏɴᴇ ᴏʀ ᴏɴᴇꜱ ᴍᴀᴋᴇ ᴛʜᴇ ᴇQᴜᴀᴛɪᴏɴ ᴀ ᴛʀᴜᴇ ꜱᴛᴀᴛᴇᴍᴇɴᴛ. ꜰᴏʀ ᴇxᴀᴍᴘʟᴇ, ᴄᴏɴꜱɪᴅᴇʀ ᴛʜᴇ ᴛʀɪɢᴏɴᴏᴍᴇᴛʀɪᴄ ᴇQᴜᴀᴛɪᴏɴ: 2 ꜱɪɴ θ = 1, ꜰᴏʀ 0° ≤ θ ≤ 360°. ꜱᴏʟᴠɪɴɢ, ᴡᴇ ʜᴀᴠᴇ ꜱɪɴ θ = 1/2. ᴛʜᴇ ᴏɴʟʏ ᴠᴀʟᴜᴇꜱ ᴏꜰ ᴛʜᴇ ᴠᴀʀɪᴀʙʟᴇ θ ʙᴇᴛᴡᴇᴇɴ 0° ᴀɴᴅ 360° ᴡʜɪᴄʜ ᴍᴀᴋᴇ ᴛʜɪꜱ ᴛʀɪɢᴏɴᴏᴍᴇᴛʀɪᴄ ᴇQᴜᴀᴛɪᴏɴ ᴀ ᴛʀᴜᴇ ꜱᴛᴀᴛᴇᴍᴇɴᴛ ᴀʀᴇ 30° ᴀɴᴅ 150°.

ᴏɴ ᴛʜᴇ ᴏᴛʜᴇʀ ʜᴀɴᴅ, ᴀ ᴛʀɪɢᴏɴᴏᴍᴇᴛʀɪᴄ ɪᴅᴇɴᴛɪᴛʏ ɪꜱ ᴀ ꜱᴘᴇᴄɪᴀʟ ᴛʏᴘᴇ ᴏꜰ ᴛʀɪɢᴏɴᴏᴍᴇᴛʀɪᴄ ᴇQᴜᴀᴛɪᴏɴ. ᴀ ᴛʀɪɢᴏɴᴏᴍᴇᴛʀɪᴄ ɪᴅᴇɴᴛɪᴛʏ ɪꜱ ᴀɴ ᴇQᴜᴀᴛɪᴏɴ ᴛʜᴀᴛ ɪꜱ ᴛʀᴜᴇ ꜰᴏʀ ᴀʟʟ ᴠᴀʟᴜᴇꜱ ᴏꜰ ᴛʜᴇ ᴠᴀʀɪᴀʙʟᴇ ꜰᴏʀ ᴡʜɪᴄʜ ʙᴏᴛʜ ꜱɪᴅᴇꜱ ᴏꜰ ᴛʜᴇ ᴇQᴜᴀᴛɪᴏɴ ᴀʀᴇ ᴅᴇꜰɪɴᴇᴅ.

ꜰᴏʀ ᴇxᴀᴍᴘʟᴇ, ᴄᴏɴꜱɪᴅᴇʀ ᴛʜᴇ ᴛʀɪɢᴏɴᴏᴍᴇᴛʀɪᴄ ɪᴅᴇɴᴛɪᴛʏ: ᴛᴀɴ θ = ꜱɪɴ θ/ᴄᴏꜱ θ, ꜰᴏʀ 0° ≤ θ ≤ 360°. ᴛʜɪꜱ ɪᴅᴇɴᴛɪᴛʏ ɪꜱ ᴛʀᴜᴇ ꜰᴏʀ ᴀʟʟ ᴠᴀʟᴜᴇꜱ ᴏꜰ ᴀɴɢʟᴇ θ ʙᴇᴛᴡᴇᴇɴ ᴀɴᴅ ɪɴᴄʟᴜᴅɪɴɢ 0° ᴀɴᴅ 360°, ᴇxᴄᴇᴘᴛ ꜰᴏʀ θ = 90° ᴀɴᴅ θ = 270°; ᴀᴛ θ = 90° ᴀɴᴅ θ = 270°, ᴄᴏꜱ θ = 0; ᴄᴏɴꜱᴇQᴜᴇɴᴛʟʏ, ᴛᴀɴ 90° = ꜱɪɴ 90°/ᴄᴏꜱ 90°= 1/0 ɪꜱ ᴜɴᴅᴇꜰɪɴᴇᴅ, ᴀɴᴅ ᴛᴀɴ 270° = ꜱɪɴ 270°/ᴄᴏꜱ 270° = ‒1/0 ɪꜱ ᴜɴᴅᴇꜰɪɴᴇᴅ!

ᴀꜱ ᴀɴᴏᴛʜᴇʀ ᴇxᴀᴍᴘʟᴇ, ᴄᴏɴꜱɪᴅᴇʀ ᴛʜᴇ ɪᴅᴇɴᴛɪᴛʏ: ꜱɪɴ² θ + ᴄᴏꜱ² θ = 1, ꜰᴏʀ 0° ≤ θ ≤ 360°. ᴛʜɪꜱ ɪᴅᴇɴᴛɪᴛʏ ɪꜱ ᴛʀᴜᴇ ꜰᴏʀ ᴀʟʟ ᴠᴀʟᴜᴇꜱ ᴏꜰ θ ᴡɪᴛʜ ɴᴏ ʀᴇꜱᴛʀɪᴄᴛɪᴏɴꜱ ᴏɴ θ! ɪꜰ, ꜰᴏʀ ᴇxᴀᴍᴘʟᴇ, θ = 30°, ᴛʜᴇɴ, ᴡᴇ ʜᴀᴠᴇ:

ꜱɪɴ²(30°) + ᴄᴏꜱ²(30°) = 1

(ꜱɪɴ 30°)² + (ᴄᴏꜱ 30°)² = 1

(1/2)² + (√3/2)² = 1

(1/4) + (3/4) = 1

4/4 = 1

1=1