Topic: Trigonometry
Can we evaluate trigonometric values such as [tex]\sin15^{\circ}[/tex], [tex]\cos5^{\circ}[/tex] and [tex]\tan3^{\circ}[/tex]? If yes, how?
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Topic: Trigonometry
Can we evaluate trigonometric values such as [tex]\sin15^{\circ}[/tex], [tex]\cos5^{\circ}[/tex] and [tex]\tan3^{\circ}[/tex]? If yes, how?
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★ Concept :-
In order to solve this question we need some pre requisite knowledge about some angles and there designated values. Actually the value of some angles when applied with Trigonometric ratios, are found using different tables. They are very complicated to be found using sums. So we need to know those and when it comes to exams, values of such angles will be given or the table of values will be provided.
Here we shall firstly apply Trigonometric formulads to convert the values of some angles. Then apply their values to find the final answer.
Let's do it !!
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★ Solution :-
• sin 15° ::
Yes we can calculate the value of sin 15°.
We know that 45° - 30° = 15°
So the above angle can be written as,
→ sin 15° = sin(45° - 30°)
We know that,
Here A = 45° and B = 30°
By applying this formula in the above relation, we get
→ sin 15° = sin 45° cos 30° - cos 45° sin 30°
Also we know that,
» sin 45° = 1/√2
» sin 30° = 1/2
» cos 45° = 1/√2
» cos 30° = 3/√2
Now applying these values in the above equation, we get
Since this cannot be further simplified. So it is the required answer.
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• cos 5° ::
By convention we know that,
→ For acute angles when θ is small then sin θ ≈ θ in radians.
Here the angle is in degree. And we know that,
→ 1° = 17 milliradians
→ 1° = 0.017
Now applying sin 5° we get,
→ sin 5° = 5 × 0.017 = 0.085
(From above since sin θ ≈ θ in radians)
Now we know that,
So cos 5° can be written as,
→ sin² 5° + cos² 5° = 1
→ cos² 5° = 1 - sin² 5°
Now by applying the value of sin 5° we get,
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• tan 3° ::
Actually the value of tan 3° cannot be found directly unless we know value of some angles.
Here 3° is an acute angle. We know that,
By applying this here, we get
→ tan 3° = cot(90° - 3°)
→ tan 3° = cot(87°)
→ tan 3° = cot 87°
Now we know that,
>> cot θ = 1/tan θ
By applying this for cot 87° , we get
By applying this here, we get
Now we know that,
→ tan 3° = cot 87°
By applying this, we get
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★ More to know :-
• sin θ = cos(90° - θ)
• cosec θ = sec(90° - θ)
• sec² θ = 1 + tan² θ
• cosec² θ = 1 + cot² θ
Answer:
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