Video Transcript
Find the value of three multiplied
by the sin of 30 degrees multiplied by the sin of 60 degrees minus the cos of zero
degrees multiplied by the sec of 60 degrees plus sin of 270 degrees multiplied by
cos squared of 45 degrees.
In order to answer this question,
we will need to recall the sine and cosine of our special angles 30, 45, and 60
degrees. We will also need to recall the
reciprocal trigonometric identities, specifically the secant of any angle. Finally, we’ll need to find the
product of two trigonometric functions as well as squaring the cos of 45
degrees.
The sin, cos, and tan of 30
degrees, 45 degrees, and 60 degrees can be set out in a table as shown. The sine of these three angles are
equal to one-half, root two over two, and root three over two, respectively. The cos of 30, 45, and 60 degrees
are equal to root three over two, root two over two, and one-half. Finally, the tangent of the three
angles equals one over root three, one, and root three. We can substitute the values of sin
of 30 degrees, sin of 60 degrees, and the cos of 45 degrees directly into our
expression. The first term three multiplied by
the sin of 30 degrees multiplied by the sin of 60 degrees is equal to three
multiplied by a half multiplied by root three over two. This is equal to three root three
over four.
From the graphs of 𝑦 equals sin 𝜃
and 𝑦 equals cos 𝜃, we see that the cos of zero degrees is equal to one and the
sin of 270 degrees is equal to negative one. The sec of any angle 𝜃 is the
reciprocal of the cos of the angle such that the sec of 𝜃 is equal to one over the
cos of 𝜃. This means that the sec of 60
degrees is equal to one over the cos of 60 degrees. As the cos of 60 degrees is equal
to one-half, the sec of 60 degrees is equal to two.
The second term in our expression,
the cos of zero degrees multiplied by the sec of 60 degrees, is therefore equal to
one multiplied by two. This is equal to two. Finally, the third term, the sin of
270 degrees multiplied by cos squared 45 degrees, is equal to negative one
multiplied by root two over two squared. Root two over two squared is equal
to two over four, as we simply square the numerator and denominator. This simplifies to one-half, which
we need to multiply by negative one. The third term is, therefore, equal
to negative one-half.
Substituting our three values into
the original expression, we have three root three over four minus two plus negative
one-half. Negative two plus negative one-half
is the same as negative two and a half, which is equal to negative five over
two. We can then multiply the numerator
and denominator of our second fraction by two. This gives us three root three over
four minus 10 over four. As the denominators are the same,
we can subtract the numerators, giving us negative 10 plus three root three over
four. This is the value of the original
expression.
