FIND THE MISSIND SIDE USING TRIGONOMETRIC RATIOS

Consider a right-angled triangle, right-angled at B.

The trigonometry ratios for a specific angle θ. There are six trigonometric ratios. 

Find the value of x, giving your answer correct to 2 decimal places :

Example 1 :

Solution :

In the figure,

Opposite side (AB)  =  x cm

Hypotenuse (AC)  =  6 cm

Here θ  =  32˚

The sides opposite and hypotenuse are involving in the trigonometric ratio sin θ

sin θ  =  Opposite side/Hypotenuse  =  AB/AC

sin 32˚  =  x/6

By using the calculator, we get

0.529  =  x/6

(0.53 × 6)  =  x

x  =  3.18

So, the value of x is 3.18 cm

Example 2 :

Solution :

In the figure,

Hypotenuse side AC  =  x cm

Adjacent side AB  =  5 cm

Here θ  =  46˚

The sides hypotenuse and adjacent are involving in the trigonometric ratio sec θ

sec 46˚  =  Hypotenuse/Adjacent side  =  AC/AB

sec 46˚  =  x/5

By using the calculator, we get

1.44  =  x/5

(1.44 × 5)  =  x

x  =  7.2

So, the value of x is 7.20 cm

Example 3 :

Solution :

In the figure,

Opposite side AB  =  5 cm

Adjacent side BC  =  x cm

Here θ  =  28˚

The sides opposite and adjacent are involving in the trigonometric ratio tan θ

tan 28˚  =  Opposite side/Adjacent side  =  AB/BC

tan 28˚  =  5/x

By using the calculator, we get

0.532  =  5/x

x  =  5/0.532

x  =  9.4

So, the value of x is 9.40 cm

Find, to 2 significant figures, the value of θ :

Example 4 :

Solution  :

In the figure,

Opposite side AB  =  4 cm

Hypotenuse side AC  =  6 cm

Here θ  =  θ˚

The sides opposite and hypotenuse are involving in the trigonometric ratio sin θ

sin θ  =  Opposite side/Hypotenuse side  =  AB/AC

sin θ˚  =  4/6

sin θ˚  =  2/3

 θ˚  =  sin^-1(2/3)

By using the calculator, we get

θ  =  42˚

So, the value of θ is 42˚

Example 5  :

Solution :

In the figure,

Hypotenuse side AC  =  7 cm

Adjacent side BC  =  5 cm

Here θ  =  θ˚

The sides hypotenuse and adjacent are involving in the trigonometric ratio sec θ

sec θ˚  =  Hypotenuse/Adjacent side  =  AC/BC

sec θ˚  =  7/5

θ˚  =  sec^-1 (7/5)

By using the calculator, we get

θ  =  44˚

So, the value of θ is 44˚

Example 6 :

Solution :

In the figure,

Hypotenuse side AC  =  7.2 cm

Adjacent side AB  =  5 cm

Here θ  =  θ˚

The sides hypotenuse and adjacent are involving in the trigonometric ratio sec θ

sec θ˚  =  Hypotenuse/Adjacent side  =  AC/AB

sec θ˚  =  7.2/5

θ˚  =  sec^-1 (7.2/5)

By using the calculator, we get

θ  =  46˚

So, the value of θ is 46˚

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