Video: Evaluating Trigonometric Functions Using Pythagorean Identities

In this video, we will learn how to use the Pythagorean identities to find the values of trigonometric functions.

15:16

Video Transcript

In this video, we will learn how to use the Pythagorean identities to find the values of trigonometric functions. We will begin by recalling some key formulae and definitions.

We know that the trig functions are periodic and have an infinite number of solutions. However, in this video, we will focus on solutions between zero and 360 degrees using the CAST diagram. The 𝑥- and 𝑦-axes create four quadrants as shown. Quadrant one has positive 𝑥- and positive 𝑦-values; quadrant two has negative 𝑥- and positive 𝑦-values; quadrant three, negative 𝑥 and negative 𝑦; and quadrant four, positive 𝑥 and negative 𝑦.

When dealing with our trig functions, we measure in a counterclockwise direction from the positive 𝑥-axis. This means that quadrant one contains angles between zero and 90 degrees; quadrant two, between 90 and 180 degrees; and so on. We can label the quadrants with the acronym CAST as shown. In quadrant one, labeled A, the sin of angle 𝜃, cos of angle 𝜃, and tan of angle 𝜃 are all positive. In quadrant two, the sin of angle 𝜃 is positive. However, the cos of angle 𝜃 and tan of angle 𝜃 are both negative. In quadrant three, when 𝜃 lies between 180 and 270 degrees, the tan of 𝜃 is positive, whereas the sin of 𝜃 and cos of 𝜃 are both negative. Finally, in quadrant four, the cos of 𝜃 is positive and the sin and tan of 𝜃 are negative.

In this video, we will also need to recall the three reciprocal trig identities. The csc of angle 𝜃 is equal to one over the sin of angle 𝜃. The sec of angle 𝜃 is equal to one over the cos of angle 𝜃. And the cot of angle 𝜃 is equal to one over the tan of angle 𝜃. If the functions sin 𝜃, cos 𝜃, and tan 𝜃 are positive or negative, then their reciprocal will also be positive or negative. This means that all six functions are positive in the first quadrant. In the second quadrant, between 90 and 180 degrees, the sin of 𝜃 and csc of 𝜃 will be positive, whereas all four other functions will be negative. This pattern continues in quadrant three and four.

In the questions that follow in this video, we will need to use this information to decide whether our answers are positive or negative.

We will now consider the three Pythagorean identities. Firstly, we have sin squared 𝜃 plus cos squared 𝜃 is equal to one. Our second identity is tan squared 𝜃 plus one is equal to sec squared 𝜃. Whilst we don’t need to consider the proofs of the identities in this video, we can get from the first identity to the second identity by dividing each term by cos squared 𝜃. sin squared 𝜃 divided by cos squared 𝜃 is tan squared 𝜃, as sin 𝜃 over cos 𝜃 equals tan 𝜃. Dividing cos squared 𝜃 by cos squared 𝜃 gives us one. From the reciprocal identities, we know that one divided by cos squared 𝜃 is sec squared 𝜃. The third Pythagorean identity is one plus cot squared 𝜃 is equal to csc squared 𝜃. This can be obtained by dividing each term of the first identity by sin squared 𝜃.

sin squared 𝜃 divided by sin squared 𝜃 is equal to one. cos squared 𝜃 divided by sin squared 𝜃 is equal to one over tan squared 𝜃. And this is equal to cot squared 𝜃. Finally, one divided by sin squared 𝜃 is equal to csc squared 𝜃. We will now use these three identities together with the CAST diagram to solve some trig equations.

Find cos 𝜃 given sin 𝜃 is equal to negative three-fifths, where 𝜃 is greater than or equal to 270 degrees and less than 360 degrees.

There are lots of ways of solving this problem. In this video, we will use the Pythagorean identity sin squared 𝜃 plus cos squared 𝜃 is equal to one. Before substituting in our value of sin 𝜃, it is worth noting that 𝜃 must be greater than or equal to 270 degrees and less than 360 degrees. Using our knowledge of the CAST diagram, this means that 𝜃 must lie in the fourth quadrant. In this quadrant, the cos of angle 𝜃 is positive, whereas the sin of 𝜃 and tan of 𝜃 are negative. This ties in with the fact that we are told that sin 𝜃 is negative three-fifths. We know that our answer for cos of 𝜃 must be positive.

We can now substitute the value of sin 𝜃 into our Pythagorean identity. This gives us negative three-fifths squared plus cos squared 𝜃 is equal to one. Squaring a negative number gives a positive answer. Therefore, negative three-fifths squared is equal to nine twenty-fifths. We can then subtract this from both sides of our equation. cos squared 𝜃 is equal to 16 over 25 or sixteen twenty-fifths. Square rooting both sides of this equation, we see that cos of 𝜃 is equal to positive or negative the square root of 16 over 25. When square rooting a fraction, we simply square root the numerator and denominator separately. The square root of 16 is four, and the square root of 25 is five. As the cos of 𝜃 must be positive, we can conclude that the cos of 𝜃 is four-fifths.

In our next question, we need to evaluate the sine function given the cosine function and the quadrant of an angle.

Find the value of sin 𝜃 given cos of 𝜃 is equal to negative 21 over 29, where 𝜃 is greater than 90 degrees and less than 180 degrees.

In order to answer this question, we’ll use the Pythagorean identity sin squared 𝜃 plus cos squared 𝜃 is equal to one. We are told that 𝜃 lies between 90 and 180 degrees, so it is worth considering our CAST diagram. The angle lies in the second quadrant, which means that the sin of angle 𝜃 must be positive. The cos of angle 𝜃 and tan of angle 𝜃 must be negative. This ties in with the fact that we are told that the cos of 𝜃 is equal to negative 21 over 29. It also helps us in that we know the answer for sin 𝜃 must be positive.

We can substitute the value of cos 𝜃 into the Pythagorean identity. Squaring negative 21 over 29 gives us 441 over 841. We can then subtract this from both sides of our equation such that sin squared 𝜃 is equal to one minus 441 over 841. The right-hand side simplifies to 400 over 841. We can then square root both sides of our equation so that sin of 𝜃 is equal to positive or negative the square root of 400 over 841.

Square rooting the numerator gives us 20 and the denominator 29. As sin of 𝜃 must be positive, if the cos of 𝜃 is negative 21 over 29 and 𝜃 lies between 90 and 180 degrees, then the sin of 𝜃 is equal to 20 over 29.

In our next question, we will use the Pythagorean identities to evaluate an expression.

Find the value of sin 𝜃 cos 𝜃 given sin 𝜃 plus cos 𝜃 is equal to five-quarters.

In order to solve this problem, we will need to recall the Pythagorean identities. We will begin with the equation sin 𝜃 plus cos 𝜃 is equal to five over four. We can square both sides of this equation. The left-hand side becomes sin 𝜃 plus cos 𝜃 multiplied by sin 𝜃 plus cos 𝜃. The right-hand side is equal to 25 over 16 as we simply square the numerator and denominator separately.

Distributing the parentheses or expanding the brackets using the FOIL methods, we get sin squared 𝜃 plus sin 𝜃 cos 𝜃 plus sin 𝜃 cos 𝜃 plus cos squared 𝜃. We can group or collect the middle terms. sin squared 𝜃 plus two sin 𝜃 cos 𝜃 plus cos squared 𝜃 is equal to 25 over 16. One of the Pythagorean identities states that sin squared 𝜃 plus cos squared 𝜃 is equal to one. This means we can rewrite the left-hand side of our equation as two sin 𝜃 cos 𝜃 plus one. We can then subtract one from both sides of this equation. 25 over 16 minus one is equal to nine over 16.

We can then divide both sides of this new equation by two, giving us sin 𝜃 cos 𝜃 is equal to nine over 32. If sin 𝜃 plus cos 𝜃 is equal to five over four, then sin 𝜃 multiplied by cos 𝜃 is equal to nine over 32.

In our final question, we will need to use a different Pythagorean identity.

Find the sec of 𝜃 minus the tan of 𝜃 given the sec of 𝜃 plus the tan of 𝜃 is equal to negative 14 over 27.

We recall that the difference of two squares states that 𝑥 squared minus 𝑦 squared is equal to 𝑥 plus 𝑦 multiplied by 𝑥 minus 𝑦. This means that sec 𝜃 plus tan 𝜃 multiplied by sec 𝜃 minus tan 𝜃 is equal to sec squared 𝜃 minus tan squared 𝜃. We are told in the question the value of sec 𝜃 plus tan 𝜃. It is equal to negative 14 over 27. And we need to calculate the value of sec 𝜃 minus tan 𝜃.

One of the Pythagorean identities states that tan squared 𝜃 plus one is equal to sec squared 𝜃. If we subtract tan squared 𝜃 from both sides of this equation, we have sec squared 𝜃 minus tan squared 𝜃 is equal to one. This is equivalent to the right-hand side of our equation. Negative 14 over 27 multiplied by sec 𝜃 minus tan 𝜃 is equal to one. We can then divide both sides of this equation by negative 14 over 27. We know that dividing by a fraction is the same as multiplying by the reciprocal of this fraction. Therefore, the right-hand side is equal to one multiplied by negative 27 over 14.

If the sec of 𝜃 plus the tan of 𝜃 is equal to negative 14 over 27, then the sec of 𝜃 minus the tan of 𝜃 is equal to negative 27 over 14. These values are the reciprocal of one another.

We will now summarize the key points from this video. The three Pythagorean identities are as follows. sin squared 𝜃 plus cos squared 𝜃 is equal to one. tan squared 𝜃 plus one is equal to sec squared 𝜃. One plus cot squared 𝜃 is equal to csc squared 𝜃. We have seen in this video that we can use these identities to find the values of trig functions. When solving any trig equations, it is also important to recall which of the functions are positive and negative in each quadrant. One way of doing this is using the CAST diagram as shown.

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