Should i drop trigonometry

We can now use the sine rule to solve simple surveying problems involving non-right-angled triangles. Find how far each point is from the house, correct to the nearest metre. We draw a diagram to represent the information. We can find the angles in AHB. Apply the sine rule to ABH :. Thus B is approximately metres from the house. Thus A is approximately metres from the house. Draw a diagram and then use the sine rule to find the distance AP and hence the exact height of the building.

Finally, evaluate the height OA to the nearest centimetre. The sine rule can be used to find angles as well as sides in a triangle. One of the known sides, however, must be opposite one of the known angles. Assuming that all the angles are acute. Both the sine rule and the cosine rule are used to find angles and sides in triangles. What happens when one of the angles is obtuse? To deal with this we need to extend the definition of the basic trigonometric ratios from acute to obtuse angles.

We use coordinate geometry to motivate the extended definitions as follows. We draw the unit circle centre the origin in the Cartesian plane and mark the point on the circle in the first quadrant.

We must show that the two definitions agree. Triangle OPQ has its vertex P on the unit circle. Draw a diagram showing the point on the unit circle at each of the above angles.

Use the coordinates of to complete the entries in the table below. Hence the tangent of an obtuse angle is the negative of the tangent of its supplement.

In our work on congruence, it was emphasized that when applying the SAS congruence test, the angle in question had to be the angle included between the two sides.

Then the angle opposite PQ is not uniquely determined. There are two non-congruent triangles that satisfy the given data. This situation is sometimes referred to as the ambiguous case. We know from the SAS congruence test, that a triangle is completely determined if we are given two sides and the included angle. However, if we know two sides and the included angle in a triangle, the sine rule does not help us determine the remaining side.

The second important formula for general triangles is the cosine rule. Drop a perpendicular from B to AC and mark the lengths as shown in the diagram. This last formula is known as the cosine rule. The cosine rule is also true when C is obtuse, but note that in this case the final term in the formula will produce a positive number, because the cosine of an obtuse angle is negative.

Some care must be taken in this instance. Applying the cosine rule:. Prove that the cosine rule also holds in the case when C is obtuse. We know from the SSS congruence test that if the three sides of a triangle are known then the three angles are uniquely determined. Again, the sine rule is of no help in finding them since it requires the knowledge of at least one angle, but we can use the cosine rule instead.

Students may care to rearrange the cosine rule or learn a further formula. Using this form of the cosine rule often reduces arithmetical errors. A triangle has side lengths 6 cm, 8 cm and 11 cm.

Find the smallest angle in the triangle. The smallest angle in the triangle is opposite the smallest side. Extension — The longest side and the largest angle of a triangle. But I think we can all agree that it should come back, if only for the "awesome" joke I came up with as I was falling asleep last night: Haversine?

I don't even know 'er! You've been warned. In the table of secret trig functions, "ha" clearly means half; the value of haversine is half of the value of versine, for example.

Complementary angles add up to 90 degrees. In a right triangle, the two non-right angles are complementary. For instance, the cosine of an angle is also the sine of the complementary angle.

Likewise, the coversine is the versine of the complementary angle, as you can see in light blue above one of the red sines in the diagram at the top of the post. The one bonus trig function that confuses me a little bit is the vercosine. If the "co" in that definition meant the complementary angle, then vercosine would be the same as coversine, which it isn't.

Instead, the vercosine is the versine of the supplementary angle supplementary angles add up to degrees , not the complementary one. My guess is that vercosine was a later term, an analogy of the square of sine definition of versine using cosine instead. If you're a trigonometry history buff and you have more information, please let me know! In any case, the table of super-secret bonus trig functions is a fun exercise in figuring out what prefixes mean.

The views expressed are those of the author s and are not necessarily those of Scientific American. Follow Evelyn Lamb on Twitter. Already a subscriber? Sign in. Thanks for reading Scientific American. Create your free account or Sign in to continue. See Subscription Options. Discover World-Changing Science. It's well known that you can shake a stick at a maximum of 8 trig functions. The familiar sine, cosine, and tangent are in red, blue, and, well, tan, respectively.

The versine is in green next to the cosine, and the exsecant is in pink to the right of the versine. Excosecant and coversine are also in the image.

Not pictured: vercosine, covercosine, and haver-anything. Check out the Online Education website for more information on a variety of topics that can help you be a successful online student such as: exam proctoring, learning styles, computer skills, and tips for student success.

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