Video: Using Periodic Identities to Evaluate a Trigonometric Expression Involving Special Angles

Find the value of cos 135° + tan 135° + cosec 225° + cos 225°.

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Video Transcript

Find the value of cos of 135 degrees plus tan of 135 degrees plus cosec of 225 degrees plus cos of 225 degrees.

In order to answer this question, we need to recall some of our special angle properties. The sin of 45 degrees is equal to root two over two. Likewise, the cos of 45 degrees is equal to root two over two. The tan of 45 degrees is equal to one.

Next, we need to consider the graphs between zero and 360 degrees. The graph of sin 𝜃 has a maximum value of one and a minimum value of negative one. It starts at zero and ends at zero. The portion of the graph between zero and 180 degrees is symmetrical about the line 𝜃 equals 90 degrees. 45 degrees is halfway between zero and 90 degrees. 225 degrees is halfway between 180 and 270 degrees.

This means that the sin of 225 degrees is equal to the negative of sin 45 degrees. Sin of 225 degrees is, therefore, equal to negative root two over two. We can use this to calculate cosec of 225 degrees, as cosec 𝜃 is equal to one over sin 𝜃.

The cosine graph also has a maximum value of one and a minimum value of negative one. Cos of zero is equal to one. Cos of 360 is equal to one. The portion of the graph shown is symmetrical about the line 𝜃 equals 180 degrees. Once again, we have values halfway between 90 and 180, and 180 and 270. We can see that the value of cos of 135 degrees will be equal to cos of 225 degrees. This will be equal to the negative of cos 45 degrees. Once again, both of these values will be equal to negative root two over two.

Finally, we need to consider the graph of tan 𝜃, which has asymptotes at 𝜃 equals 90 and 𝜃 equals 270 degrees. There are no maximum or minimum values of tan 𝜃. The graph of tan 𝜃 is as shown. Once again, due to symmetry, we can see that tan of 135 degrees is equal to negative tan of 45 degrees. As tan 45 was equal to one, negative tan 45 must be equal to negative one.

We mentioned earlier that cosec 𝜃 was equal to one over sin 𝜃. Therefore, cosec of 225 degrees must be equal to one over sin of 225 degrees. Cosec 225 is equal to one divided by negative root two over two. This is equal to negative two over root two. We can rationalise this fraction by multiplying the numerator and denominator by root two. This is equal to negative two root two over two, which in turn simplifies to negative root two.

We will now clear some space and summarise the answers. Cos of 135 degrees was equal to negative root two over two. Tan of 135 degrees was equal to negative one. Cosec of 225 degrees was equal to negative root two. And finally, cos, or cosine, of 225 degrees was equal to negative root two over two. We need to add all four of these answers.

This gives us negative root two over two minus one minus root two minus root two over two. The two negative root twos over two will simplify to negative root two. This gives us negative root two minus one minus root two. Negative root two minus root two is equal to negative two root two. Therefore, our final answer is negative two root two minus one. The value of cos 135 plus tan 135 plus cosec 225 plus cos of 225 is negative two root two minus one.

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