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In this video, we’ll see how we can find the area of a triangle using trigonometry.
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To do this, we’ll need to know the lengths of two sides and their included angle.
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But before we do that, let’s consider a rectangle.
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We know that a rectangle has four 90-degree angles.
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It has a length and a width.
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And the area is equal to the length times the width.
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If we draw a diagonal across this rectangle, we’ve divided it in half.
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And each of these triangles have an area of one-half the length times the width.
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Let’s look more closely at the right triangle.
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When we’re dealing with the triangle, that width becomes a height.
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And instead of calling it the length, we call it the base.
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And so, we say that the area of a triangle is one-half the height times the base.
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The key to using this formula is that the height is perpendicular to the base.
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If we’re not given a perpendicular height, we can’t use this formula.
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Let’s say we’re dealing with a triangle that looks like this.
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We could call this side the base.
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But the height of this triangle is the perpendicular distance from the base to the vertex opposite that base.
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This distance is the height.
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It’s still true that the area of this triangle is one-half times the height times the base.
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That formula is true because every triangle is half of a rectangle.
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If we double this triangle and move the pieces around, we have a rectangle.
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This proves that the formula area equals one-half times height times base will find the area of any triangle as long as you have the perpendicular height.
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But this video is going to help us learn how to solve the area of triangles when we aren’t given the height.
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In these cases, we need trigonometry to come to the rescue.
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Okay, let’s go back.
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Area equals one-half the height times the base.
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And we know that the height is a perpendicular intersection of the base.
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Now, in this triangle, our base is 𝑎.
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So, let’s just leave that there.
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We want to use some trigonometry to help us find the height.
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The height of this triangle actually makes the larger triangle divided into two smaller triangles.
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And the two smaller triangles that make up the larger triangle are both right-angled triangles.
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If we call the point where the height intersects the base point 𝑃, we could say triangle 𝐴𝑃𝐶 is a right triangle.
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Let’s just move that information up a bit.
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We can also say that the line segment 𝐴𝑃 is the height of the larger triangle 𝐴𝐵𝐶.
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We can say that side 𝑏 is the hypotenuse of the smaller triangle, 𝐴𝑃𝐶.
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And now, let’s consider angle 𝐶.
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Opposite angle 𝐶 is the line segment 𝐴𝑃 or what we’ve labelled as ℎ.
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And we’ve already said that side 𝑏 is the hypotenuse.
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If we know an opposite and a hypotenuse in a right triangle, what trig value should we be thinking of?
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We can choose from the sine, cosine, or tangent relationships.
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But the sine relationship is the one that is the opposite over the hypotenuse.
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This means we can say that sin of angle 𝜃 equals ℎ over 𝑏.
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This means that the height over the side length 𝑏 will be equal to the sin of the angle 𝜃.
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We’re trying to solve for the height.
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If we multiply both sides of this equation by the side length 𝑏, then we can say that the height is equal to the sin of 𝜃 times 𝑏.
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And in case of this triangle, we know that 𝜃 is the sin of angle 𝐶.
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We can now substitute in for the height, sin of 𝐶 times 𝑏.
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Another way to find the area of a triangle is one-half times sin 𝐶 times 𝑏 times 𝑎.
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We can rearrange it to its more common form, the area of a triangle is found by one-half times 𝑎 times 𝑏 times sin of 𝐶.
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Before we move on, let’s just walk through this one more time.
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For our triangle, 𝐴𝐵𝐶, we can find its area if we know the lengths of two sides and the value of its included angle.
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If we knew side length 𝑎 and 𝑏 and the value of angle 𝐴, we could not use this rule.
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Nor could we use this rule if we knew the value of side length 𝑎 and side length 𝑏 and the angle 𝐵.
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We must know the included angle, the angle between the two side lengths we know.
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To use this formula, we must have two side lengths and an included angle.
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It does not matter which two sides, as long as the angle we’re given is included between those two sides.
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Now, let’s move on to some examples.
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Which of the following is a formula that can be used to find the area of a triangle?
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A) one-half 𝑎𝑏 cos 𝐶, B) one-half 𝑎𝑏 sin 𝐶, C) one-third 𝑎𝑏 sin 𝐶, D) one-fourth 𝑎𝑏 cos 𝐶, or E) one-fourth 𝑎𝑏 sin 𝐶.
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If we sketch a triangle and label it 𝐴, 𝐵, and 𝐶, the side length opposite vertex 𝐴 is usually labelled with a lower case 𝑎.
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The side length opposite vertex 𝐵 is labelled with a lower case 𝑏.
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And we label lower case 𝑐 the side length opposite vertex 𝐶.
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We have to remember that a triangle is half of a rectangle.
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And so, it’s unlikely that options C through E would be the answer.
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We noticed that options A and B are dealing with the angle at vertex 𝐶, that’s this angle, and the lengths 𝑎 and 𝑏.
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At this point, we recognize that we have two sides and an included angle.
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And we know that the height of this triangle will be equal to 𝑏 times the sin of 𝐶.
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To use trigonometry to solve for the area of a triangle, we take one-half times 𝑎 times 𝑏 times sin of 𝐶, which is option B here.
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For this example, we need to apply the formula we’ve been talking about.
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In the given figure, work out the area of the triangle to two decimal places.
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In this figure, we were given two side lengths and an included angle.
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Since we have this information, we can use the formula 𝐴 equals one-half times 𝑎𝑏 times sin of 𝐶 to find the area, where 𝑎 and 𝑏 are side lengths and 𝐶 is an included angle.
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The area of this triangle is found by one-half times 10 times seven times sin of 136 degrees.
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If you plug this into your calculator, you get 24.3130429 continuing.
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If your calculator did not give you this answer, you should check and make sure that your calculator is set to degree mode and not to radians.
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To get our final answer, though, we want it rounded to two decimal places.
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There’s a one in the second decimal place.
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To the right of that, in the third decimal place, the thousandths place, is a three.
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This means we should round down to 24.31.
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We weren’t given any units, so we can just say that the area of this triangle is 24.31 units squared.
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This example might look just as simple as the last one on the surface, but it’s going to take a few more steps to find the area.
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The figure shows a triangular field with sides 670 meters, 510 meters, and 330 meters.
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Find the area of the field giving the answer to the nearest square meter.
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At this point, we’re not given the angle measure of any of the angles inside this triangle.
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That means we can’t check for a perpendicular distance that could be the height.
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Before we do anything here, we’re going to have to find at least one of the angles.
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Because we know all three sides, there is one rule we can use here.
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We can use an application of the cosine rule, which tells us 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared minus two 𝑎𝑏 times cos of 𝐶.
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Now, we actually want to find the measure of angle 𝐶.
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We don’t know what that is.
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But we can rearrange this rule so that the cos of 𝐶 is the term that we’re looking for.
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We can subtract 𝑎 squared and 𝑏 squared from both sides of the equation.
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Then, we have the statement 𝑐 squared minus 𝑎 squared minus 𝑏 squared equals negative two 𝑎𝑏 cos of 𝐶.
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And if we divide both sides by negative two 𝑎𝑏, we can say that cos of 𝐶 is equal to 𝑐 squared minus 𝑎 squared minus 𝑏 squared over negative two 𝑎𝑏.
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Our goal is to find 𝐶.
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And the 𝑏 side is equal to 510 meters.
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And the 𝑎 side is equal to 670 meters.
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And our third side is 𝑐.
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𝑐 squared is 330 squared minus 670 squared minus 510 squared over negative two times 670 times 510.
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All of this will equal the cos of angle 𝐶.
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If you enter all of that into your calculator, you will get 0.878109453 continuing is equal to the cos of angle 𝐶.
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We need to be careful here.
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Angle 𝐶 is not equal to 0.878109453 continuing degrees.
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The cos of angle 𝐶 is equal to this decimal value.
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To find angle 𝐶, we need to take the inverse cos of 0.878109453 continuing.
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On most calculators, you can hit cos inverse of the answer, of the previous answer.
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Angle 𝐶 is equal to 28.58485793 continuing degrees.
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If your calculator did not give you this answer, you should check and make sure that you’re calculating in degrees and not in radians.
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Now that we have a value for one of the angles, we have the value of two side lengths, and an included angle.
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Which means we can use the area is equal to one-half times 𝑎 times 𝑏 times sin of 𝐶, where 𝑎 and 𝑏 are side lengths and 𝐶 is an included angle.
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To find the area of this triangle, we want to say the area is equal to one-half times 670 times 510 times the sin of the answer already in your calculator.
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Why would we do this?
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This gives us the most accurate answer before we round.
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It’s calculating the sin of 28.58485793 continuing.
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When we do that, we get 81744.85833 continuing.
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And this is the point where we want to round to the nearest square meter.
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We wanna round to the nearest whole number.
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We have a four in the ones place, and the digit to the right of that is an eight, telling us we need to round up.
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That four rounds up to a five, and everything to the left of the five stays the same.
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So, we have 81745 square meters.
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The area of this playing field is 81745 meters squared when we round to the nearest square meter.
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Let’s do a quick recap of the key points of finding the area of a triangle using trigonometry.
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The area of a triangle can be found by multiplying one-half times side length 𝑎 times side length 𝑏 times the sin of the included angle 𝐶.
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So, we say that this formula requires us to have two sides and an included angle.