Finding areas with trigonometry
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Finding Areas with Trigonometry. Objectives. I can use trigonometry to find the area of a triangle. Practice. Find the area of a regular triangle with a side length of 18.6 meters. A B C D. A. 346 m 2 B. 299.6 m 2 C. 173 m 2 D. 149.8 m 2. Next Application….

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Finding Areas with Trigonometry

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Finding areas with trigonometry

Finding Areas with Trigonometry


Objectives

Objectives

  • I can use trigonometry to find the area of a triangle.


Practice

Practice

Find the area of a regular triangle with a side length of 18.6 meters.

  • A

  • B

  • C

  • D

A.346 m2

B.299.6 m2

C.173 m2

D.149.8 m2


Next application

Next Application…

  • Area of an oblique triangle

    • Given two sides of any triangle and the measure of an angle between them

    • Use trigonometry to find its surface area

  • Recall previous formula for the area of a triangle: A = ½ bh


We will use an obtuse triangle

We will use an obtuse triangle

  • Label sides a, b, and c, opposite their corresponding angles

  • Draw a height, h, inside


Finding areas with trigonometry

Next…

  • In order to use A = ½ bh, we need b and h, but all we know are a, b, and the measure of angle C (for example) we need “h”!

  • Look at triangle BDC inside:

    • How can we write a trig ratio

      using sides h and a?

    • We can use this to

      solve for “h”!


So far we have

So Far we have…

  • Solve this for “h”: h = a sin C

  • Now we have the info we need to use A = 1/2bh!

  • A = ½ bh substitute “a sin C” for “h”

  • A = ½ a b sin C


In conclusion

IN CONCLUSION

  • The area of an oblique triangle is one-half the product of the lengths of two sides, times the sine of their included angle!

  • For any triangle, ABC

    Area = ½ bcsinA = ½ absinC = ½ ac sinB


Practice1

Practice

  • Find the area of a triangular lot having two sides of lengths 90m and 52m and an included angle of 102°.

  • Draw it:

  • Area = ½ (90)(52) sin 102

    ≈ 2288.87 m2


Practice2

Practice

  • Find the area of a triangle with sides 6 and 10 and an included angle of 110° Round to the nearest hundredth.

  • Area = 28.19


Practice3

Practice

  • Find the area of a triangle with side lengths 92 and 30 with an included angle 130°.

  • Area = 1057.14


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