evaluate each of the following...sin 75-cos 105-sin 15+cos 15
I assume you want "exact" answers for these, the usual kind of request for these types of questions.
All your given angles are combinations of the standard angles of 30, 45, 60 etc
e.g. 75 = 45+30
15 = 45-30
105 = 60+45
You should also have memorized or have quick access to the trig ratios of these
e.g. sin 45° = √2/2 , cos 60° = 1/2 , etc
evaluate each one, the do the arithmetic
e.g.
sin75 = sin(45+30)
= sin45cos30 + cos45sin30
= √2/2(√3/2) + √2/2(1/2)
= √6/4 + √2/4 = (√6 + √2)/4
carefully do the others, easy to make errors with those radicals
You could also have done some preliminaray simplification using the complementary angle properties
e.g.
cos15° = sin75°
cos105 = sin(-15) = -sin15
so sin 75-cos 105-sin 15+cos 15
sin75 - (-sin15) + sin15 + sin75
= 2sin75 + 2sin15
= 2(sin75+sin15)
Now you only have to do two of the calculations
To evaluate the expression sin 75 - cos 105 - sin 15 + cos 15, we can use the trigonometric identities and a scientific calculator. Here's how you can solve it step by step:
1. Start by evaluating sin 75. To do this, use your scientific calculator and enter the angle in degrees: sin(75). The result for sin 75 is approximately 0.9659.
2. Next, evaluate cos 105. Similarly, use your calculator and enter the angle in degrees: cos(105). The result for cos 105 is approximately -0.2588.
3. Now, calculate sin 15. Use your calculator and enter sin(15). The result for sin 15 is approximately 0.2572.
4. Lastly, evaluate cos 15. Use your calculator and enter cos(15). The result for cos 15 is approximately 0.9659.
Now, substitute the values we obtained into the expression:
sin 75 - cos 105 - sin 15 + cos 15 = 0.9659 - (-0.2588) - 0.2572 + 0.9659.
Simplifying further:
0.9659 + 0.2588 - 0.2572 + 0.9659 = 1.9326
So, the value of the expression sin 75 - cos 105 - sin 15 + cos 15 is approximately 1.9326.