Question

how to calculate the length of the side x?​

Answer 1

$\large\underline{\sf{Solution-}}$

In right angle triangle ABC, right angled at C, it is given that AB = 5 cm and BC = 3 cm.

So, Using Pythagoras Theorem, we have

$\rm \: {AB}^{2} = {AC}^{2} + {BC}^{2}$

$\rm \: {5}^{2} = {AC}^{2} + {3}^{2}$

$\rm \: 25 = {AC}^{2} + 9$

$\rm \: 25 - 9= {AC}^{2}$

$\rm \: 16= {AC}^{2}$

$\rm \: {4}^{2} = {AC}^{2}$

$\rm\implies \:AC \: = \: 4 \: cm$

From figure, we concluded that AC = CD

So, it implies AD = AC + CD = 4 + 4 = 8 cm.

Now, in right angle triangle ADE, right angled at D, we have

AD = 8 cm

AE = x cm

sin m = 0.48

So,

$\rm \: sin \: m \: = \: \dfrac{AD}{AE}$

$\rm \: 0.48 \: = \: \dfrac{8}{x}$

$\rm \: 0.48x \: = \: 8$

$\rm \: x = \dfrac{8}{0.48}$

$\rm \: x = \dfrac{800}{48}$

$\rm \: x = \dfrac{50}{3} \: cm \:$

$\rule{190pt}{2pt}$

Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1

Answer 2

$\sf\fbox\red{your\:Solution}$

$\: \sf \: \: from \: pythagoras \: theorem \: \\ \\ \sf→ \: {AB}^{2} = {AC}^{2} + {BC}^{2} \\ \sf \: → {5}^{2} = {AC}^{2} + {3}^{2} \\ \sf \: →25 = {AC}^{2} + 9 \\ \sf \: → {AC}^{2} = 25 - 9 \\ \sf \: →{AC}^{2} \: = \sqrt{16} \\ \sf \: → {AC}^{2} = {4}^{2} \\ \sf \: →AC = 4$

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$\: \sf \: as \: shown \: in \: figure \: →AC=DC \\ \sf \: therefore \: we \: get \: that \\ \\ \sf \: →AD=AC+CD \\ \sf \: →AD = 4 + 4 \\ \sf→AD = 8cm$

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$\sf \: AE = X cm \\ \sf Sin \: m = 0.48 \\ \\ \sf →\frac{AD}{AE} \sf \: = Sin \: m \: \: \: \: \: \\ \sf \: → \frac{8cm}{x} = 0.48 \: \: \: \: \: \: \: \: \\ →x = \frac{0.48}{8} \times 100 \\ →x = \frac{800}{48} \times \frac{16}{16} \: \: \: \\ →x = \frac{50}{3} cm \: \: \: \: \: \: \: \: \:$

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$\sf \: therefore \: \\ \\ \sf \red{→AC = 4} \: \: \: \: \: \: \\ \sf\pink {→AD = 8 } \: \: \: \: \: \\ \sf \orange {→X = \frac{50}{3} cm}$

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$\sf\fbox\red{additional information}$

The six trigonometric ratios :-

• sine (sin)
• cosine (cos)
• tangent (tan)
• cotangent (cot)
• cosecant (cosec)
• secant (sec).

definition :-

• sine: Sine of an angle is defined as the ratio of the side opposite(perpendicular side) to that angle to the hypotenuse.

• cosine: Cosine of an angle is defined as the ratio of the side adjacent to that angle to the hypotenuse.

• tangent: Tangent of an angle is defined as the ratio of the side opposite to that angle to the side adjacent to that angle.

• cotangent: Cotangent is the multiplicative inverse of the tangent

• cosecant: Cosecant is a multiplicative inverse of sine.

• secant: Secant is a multiplicative inverse of cosine.

some value of trigonometry ratio :-

• Sin 30° = 1/2
• Cos 30° = √3/2
• tan 30° = 1/√3
• sin 60° = √3/2
• cos 45° = 1/√2
• tan 45° = 1